Transformation problem

In Marxist Economics the Transformation Problem is the problem of finding a general rule (or set of functional relationships) to "transform" Marx’s "economic values" defined and used in Capital's Volume I into the "competitive prices" (or "production prices") of Capital's Volume III.

The transformation problem comes into being when Marxist economics are used to predict opportunity costs. Within Capital Volume I states the relations of value between commodities, and the nature of the relations of production in a productive environment. Volume III describes the relationships between distributors of production. A solution to the transformation problem would unite Marx's statements regarding the structure of production with Marx's statements regarding the nature of distribution. Thus solved, an analysis of the nature of the relations of production would be able to predict the consumer prices of various commodities produced. Thus, an economist who wished to equate prices of commodities, or maximise profits, could use Marx's theory in a predictive sense to compare the opportunity costs of various production, sale and purchase options.

It has been demonstrated to the satisfaction of most economists that a general solution to the transformation problem does not exist. Two special solutions have been demonstrated. The earliest well developed criticisms of the transformation problem as insoluable were presented by 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 to prove this result in a simple and general way. The standard reference is now Samuelson (1971).

The logical proof that Marxian values have generally no relationship with competitive prices is now accepted by the majority of scholars. This proof has important implications for Marx's theory of labour exploitation and economic dynamics, two theories where the correspondance between Marx's "value" and "price" play a crucial role.

However, a number of credible Marxist scholars reject the Sraffian findings, and maintain that the transformation problem is soluable. Others reject the relevence of the transformation of value into price for the key elements of Marxian political economy. Further scholars, like Harry Cleaver reject Marxist economics outright, and emphasise the politics of the relations of production over the distributive economic function.

This article uses Adam Smith's example of labour in production to survey Marx’ labour theory of value in relation to the transformation problem.

British classical labour theory of value

Marx’ value theory was developed from a rather different idea introduced in economics by Adam Smith and used by many British classical economists.

In British classical economic thought, relative embodied labour (i.e., the relative amount of non-uniform labour normally required by the production of one unit of each good) was seen as an approximation to relative competitive prices (or natural prices), which are given by relative unit costs of production. Both Smith and Ricardo were well aware that this approximation is an exact one only in the simplest cases (and Smith in fact mentioned but did not use it).

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. Call the unit labour-input requirement for the production of each good $l_{i}$ where $l$ is labour and $i$ may be $B$ oder $D$, ie, let $l_{i}$ be the number of hours of labour to catch either a beaver or a deer. One deer worth $l_{D}$ hours would then find its value reflected in ${l_{D} \over l_{B}}$ 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 "relative price" or unit "economic value" 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 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 $P_{i}=wl_{i}$).

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 arrows’ competitive price, times the proportion $h\leq 1$ of arrows lost after each shot.

Second, there is the net rental or return required by the arrow’s owner (who might or might not be the same person as the hunter using it). This can be expressed as a (uniform) rate of return:

$rP_{A}$

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. These cases are: 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 wage 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’ transformation problem.

Marx’s labour theory of value

Unlike the British classical economists, Marx defined the intrinsic value of commodities as the total amount of uniform labour socially required for, and so embodied in, its production. Marx developed his labour theory of value in Chapter I of Capital, arguing that the value of a commodity can only find its reflection in the value of another commodity when they have something in common. Marx went on to determine that common element as labour power, or the amount of effort exerted in the production of that commodity. This made labour power the substance of value for Marx.

Surplus value and exploitation

According to Marx, labour power manifests in a number of ways. Firstly, labour power is the actual exertion of a human being in society. As a service produced in the social context of capitalism, labour power is in itself a commodity. It is produced by the consumption of goods and services with a discrete value represented as the embodyment of the value of the consumed commodities. Labour power is for Marx the value of its inputs. As a commodity, labour power is exchanged from a worker to a capitalist in the act of production. As such labour power is one of the means of production, and can be treated as a form of capital: labour power is often said to be the only property owned by workers within capitalism.

The unit "labour value" is the amount of labour power embodied in the goods that make up the real subsistence wages rate. As Marx argues in Volume I, this rate is socially determined, and not fixed by a minimum necessary for survival. The value of labour power can fall below the minimum necessary to keep workers alive, or rise to standards of living experienced in the industrialised countries in the late 20th century. A unit of "labour value" will be indicated 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_Wl_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. This property derives from the difference in the value of the subsistance of a labourer (the actual person), and the amount of labour power which can be generated by a labourer. A labourer's subsistance may be worth four hours of labour power, but the labourer is capable of exerting any number of hours of labour power. 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 power produces more than its own value (when considered as the value of subsistance of the labourer), direct-labour input is called variable capital and denoted as v. In our previous example one has v_i = l_Wl_i.

By contrast, the value of other inputs – the indirect (or “dead”) past labour embodied in used up arrows, in our example – is transmitted to the product as it stands, without additions. It is hence called constant capital and denoted as c. In our previous example one has c_i = l_Aa_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_Aa_i + l_Wl_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_Aa_i + l_i = E_i

Solving for s_i the above two relationships one has:

s_i/v_i = (1 – l_W)/l_W = σ ∀ i

This necessarily uniform ratio s_i/v_i = σ is called by Marx the rate of surplus value, and it allows to re-write Marx’ value equations as:

p_i = c_i + v_i(1 + σ) = l_Aa_i + l_W(l_i (1 + σ)

Transformation of values into prices

Like Ricardo, Marx knew that relative labour values – p_D/p_B in the above example – do not generally tally with relative competitive prices – P_D/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 well-defined arbitraging process – the 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. Marx’ attempt to give a detailed account of this process is found in chapter 9 of Volume III.

Marx’ 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’ 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_Aa_B + l_B < 1.

Table 1 – Composition of Marxian values in the deer-beaver-arrow production model

Sector	Total Constant Capital Qici	Total Variable Capital Qivi	Total Surplus Value σQivi	Unit Value ci+(1+σ)vi
B	QBlAaB	QB(lAaB+lB)lB	σQB(lAaB+lB)lB	lAaB+(1+σ)(lAaB+lB)lB
D	QDlAaD	QD(lAaB+lB)lD	σQD(lAaB+lB)lD	lAaD+(1+σ)(lAaB+lB)lD
Total			σ (lAaB+lB)(QBlB+QDlD)

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

Table 2 – Marx’ transformation formulas in the deer-beaver-arrow production model

Sector	Total Constant Capital Qici	Total Variable Capital Qivi	Redistributed Total Surplus Value rQici	Resulting Competitive Price vi+(1+r)ci
B	QBlAaB	QB(lAaB+lB)lB	rQBlAaB	(lAaB+lB)lB +(1+r)lAaB
D	QDlAaD	QD(lAaB+lB)lD	rQDlAaD	(lAaB+lB)lD +(1+r)lAaD
Total			r lA(QBaB + QDaD) = σ (lAaB+lB)(QBlB+QDlD)

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 competitive price is obtained, as the sum of constant capital, variable capital and “profit” per unit of output, in the last column of Table 2.

Marx’ error and its correction

It was however soon noticed that Marx’ formulas for competitive prices were in fact 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’ 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.

Indeed, the correct way to compute competitive (relative) prices 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 normalizing 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)/(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)/a_B.

As the reader can check, this latter result, which gives the correct competitive price of deer in units of beavers for the overly simplified model used here, is generally inconsistent with Marx’ price formulas of Table 2.

Conclusions

The 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

The are just two very special cases where this is not true. 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/a_B = l_D/a_D. In Marxian parlance, 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’ 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 the l’s and a’s – 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

All the above conclusions have ceased to be controversial since more than a generation ago. Yet, some controversy still remains on what consequences (if any) this general failure of Marx’ transformation process may have on his theory of labour exploitation, and thus his global view of capitalist societies and their “laws of motion”.

Marxist views

Economic readings of Marx’ value theory view it as an attempt to explain prices, thus the transformation problem is central to the usefulness of Marx's economics in predicting prices. Standard economic views see price determination as the form of output distribution between labour and capital, so that no positive theory of labour exploitation could ever be built without some at least implicit reference to prices. Marx’ reference was very explicit, and this is precisely why the transformation problem worried him so much.

However, political-economic readings of Capital, such as Harry Cleaver's Reading Capital Politically emphasise exploitation as a direct control of worked time, unrelated as such to price determinations. These readings are usually associated with the autonomist strand of Marxism which priviledges the point of 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.

There also are, however, competent Marxist scholars who still think that Marx’ failure on the transformation issue has no lethal consequences on his system as a whole. As it has just been seen, in a few very special cases Marx’ idea of labour as the “substance” of (exchangeable) value would not be openly at odds with the facts of market competitive equilibrium. Some Marxists have argued that such cases – though admittedly not generally observed – throw light on the “hidden” or “pure” nature of capitalist society. Thus Marx’ 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 have suggested 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.

Mainstream views

Mainstream scholars question the assumption that the basic nature of capitalist production and distribution can be gleaned from unrealistic special cases. Moreover, from the fact that in all such special cases Marx’ reasoning can be easily turned upside down, through an inverse transformation process, they argue that Marx’ 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, p.62)

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. One would thus have a “pure” state of capitalist society where Marx’ 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’ basic argument – from the “substance” of value to the concept of exploitation – is 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 and many others had indeed noticed some time ago.

References

• Marx, K. (1867) Das Kapital Volume I.
• Marx, K. (1894) Das Kapital Volume III.
• Böhm-Bawerk, E., von (1896) ‘Zum Abschluss des Marxschen Systems’ Festgabe für Karl Knies Berlin [trans. P. Sweezy ed. (1949) Karl Marx and the close of his system by Eugen Böhm-Bawerk and Böhm-Bawerk’s criticism of Marx by Rudolf Hilferding with an appendix by L. von Bortkiewicz].
• Bortkiewicz, L. von (1906) ‘Wertrechnung und Preisrechnung im Marxschen System’ Archiv für Sozialwissenschaft und Sozialpolitik 3, XXIII and XXV [trans. ‘Value and Price in the Marxian System’ International Economic Papers 1952 2 5-60].
• Sraffa, P. (1960) Production of commodities by means of commodities.
• Samuelson, P.A. (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.