Transformation problem

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The transformation problem within the labor theory of value (here abbreviated: AWT) by Karl Marx describes the question of whether a consistent formal procedure can be specified that allows the values of goods measured in socially necessary working hours to be converted into production prices.

The problem

According to the law of value , goods are on average according to their value, i.e. H. of the labor time socially necessary for their production is exchanged ( Das Kapital , Volume I):

W = c + v + m

In Das Kapital, Volume III, however, the assumption is made that the profit rate m / (c + v) achieved by the capitalists tends to equalize itself to an average level through the competition of capital across all branches.

W = c + v + m = k + m = k + p

The production prices calculated under the conditions of this redistribution of surplus value deviate systematically from the previously calculated commodity values.

Marx assumes here:

  • Let the sum of the values ​​be equal to the sum of the prices.
  • Let the sum of the added values ​​be equal to the sum of the profits.
  • The value profit rate is the same as the price profit rate.

If one now assumes several branches of production with different rates of profit, one must also take into account the interdependence of the branches of production with one another. The difficulty then lies in accomplishing a simultaneous transformation that makes the total profit equal to the total surplus value and at the same time the total prices of production equal to the total values, with the average profit rate as the invariant bracket between the two valuation systems.

When values ​​are transformed into prices, the ratio of price to value must be the same in the case in which a given commodity is regarded as input as in the case in which it is regarded as output (an exception to this assumption the Temporal Single System Interpretation of the AWT discussed later , which explicitly evaluates inputs and outputs differently). After the transformation, the rate of profit must turn out to be the same for each of the capitals in question (an exception to this assumption is again the stochastically oriented approach of Farjoun / Machover discussed later ).

The proportions of price to value and the rate of profit are considered to be the main unknowns. The transformation problem can thus be mathematically reduced to:

"Can a determined system of equations be specified for the relationships between the various branches of production and the conditions that must be met as a result of the transformation?"

(The number of independent equations must correspond to the number of unknowns!)

To the history of origin

Karl Marx , Theories of Added Value , 1956

Karl Marx wrote the three volumes of Capital and the Theories of Surplus Value in almost the reverse order from when they were published. After his death, the remaining manuscripts of Capital were edited by Friedrich Engels , the theories on surplus value (often referred to as the “fourth volume of Capital ”) by Karl Kautsky. The editor Engels had already put the problem of converting values ​​into prices up for public discussion (Preface Vol. II, Vol. III and Afterword). In the course of the decades, this resulted in a wave of proposed solutions that Engels referred to as “price riddle literature”.

The first high point was the controversy between Eugen von Böhm-Bawerk and Rudolf Hilferding , which, however, did not yet deal with the transformation problem in the narrower and mathematically formalized sense. Ladislaus von Bortkiewicz began to apply mathematical methods of linear algebra to the problem . Finally, Paul A. Samuelson took on the formal solution to the problem raised and claimed that there is no algorithm that leads from labor values ​​to production prices:

“Consider two alternative, contradicting systems. Write the one down. To transform, take an eraser and erase it. Then write down the other one instead. Voilà! This ends the transformation algorithm. "

Samuelson's methodical approach was sharply rejected from a Marxist perspective by Paul Mattick (1974) in his essay Samuelson's transformation of Marxism into bourgeois economic theory. The discussion in the wake of Samuelsons shows that Samuelson's contribution was not the “last word” on the subject that the author had hoped for, but rather the preliminary intermediate stage of a certain interpretation scheme in a debate that continues to this day.

Although Marx, like Engels, attached great importance to the solution of this problem, especially in its formal form of representation, the question arises as to why the AWT could be forced to combine the distribution of income (including the generation of surplus value) and the relative price structure in one and the same model to explain. As Pierangelo Garegnani has shown and it has been shown not least in the capital controversy , the equilibrium theory favored by Paul Samuelson as an alternative to this, together with the marginal productivity theory , is not yet able to answer the question of value determination (here: of capital).

According to critics, Marx either demonstrated a fundamental inconsistency in economic model thinking or showed that the calculation of values ​​is completely superfluous compared to the calculation in production prices. The background is that this also believes the theory of exploitation and surplus value to be dispensed with. This view remains controversial, however, because there are a number of consistent solutions to the transformation problem and therefore the claim of a “redundancy of value theory” can only be maintained if the problem is interpreted in a very specific and narrow way.

The Ladislaus von Bortkiewicz solution

The classic solution to the problem, also favored by Paul M. Sweezy , while maintaining the AWT, comes from Ladislaus von Bortkewitsch and was developed at the beginning of the 20th century. In a simultaneous system of equations, all commodity values ​​(including those parts of constant and variable capital that were not transformed in Marx) should be converted into production prices. Due to this system of simultaneous mutual interdependence, one of Marx's "invariance postulates" had to be:

A) Sum of prices = sum of values ​​or

B) Sum of surplus value = sum of profits

be abandoned. Bortkiewicz decided on A) in order not to endanger the (supposedly more important) theory of exploitation and to make gold ware the numéraire . In his system of equations, three production sectors are assumed that produce means of production (Division I), consumer goods (Division II) and luxury goods (Division III). The whole thing then has the following form:

(c1x + v1y) (1 + r) = (c1 + c2 + c3) x
(c2x + v2y) (1 + r) = (v1 + v2 + v3) y
(c3x + v3y) (1 + r) = (m1 + m 2+ m3) z

Here x, y and z are the price coefficients, which can also be understood as transformation coefficients when converting values ​​into prices. The same rate of profit for all three sectors is 1 + r. In the left-hand column, the value components of the constant and variable capital of each sector are added after multiplication by the relevant conversion factor (x or y) and then multiplied by the general rate of profit so that the transformed variable and constant capital is added to the average profit . On the right hand side of the equation, e.g. B. in department 1 the constant capital of all three sectors (c1 + c2 + c3), which represents the output of department I, added up and multiplied by x as the transformation coefficient. The equal sign guarantees that all components of constant capital in all three sectors expressed in prices are identical to the transformed total expenditure of the first sector (which produces these components). Say's theorem applies here , constancy of technology and an equal turnover time for all capital. The system is purely static and connects inputs with outputs simultaneously. Time and causality do not exist in this model world, preliminary products and end products are in a relationship of quantitative identity that is not affected by the production process.

As with sector 1, the same procedure is used for sectors 2 and 3. As we can easily see, the three equations are opposed to four unknowns: The conversion coefficients x, y and z and the rate of profit r. Bortkiewicz now closes the system by specifying z = 1. Thus, the luxury goods department 3 becomes the numéraire and the identity of value unit and price unit in the luxury goods sector is assumed as given. As a result, this method means that the surplus value corresponds to the total profit, because the added surplus value of all three sectors is multiplied by one and is therefore not changed. However, this invariance is paid for in sector 3 with the effect that the value and price sum usually diverge. Why is this so?

This is because, as in every sector, also in sector 3, the organic composition of capital is above or below the social average and consequently the price of luxury goods can also vary. This change is justified by the fact that with the value-price transformation, the surplus value is converted into profit, that the basis of assessment of profitability is no longer variable capital, but constant plus variable capital. Sweezy has this provided that gold is produced as a luxury good in Sector 3, expressed as follows:

“Since ex hypothesi the price and value of a unit of gold are numerically one, the fact that its price is 'higher' than its value can only be expressed by the fact that the average price of all other goods is lower than their average value. In other words, if the organic composition of capital in the gold industry is relatively high, the transformation from value to price will increase the purchasing power of gold. "

The deviation of the total value from the total price is thus itself the direct consequence of the normalization of the third sector, because sector 3 is ultimately transformed, namely relative to the other sectors via its own purchasing power in the price system. This system does not accumulate, but consumes the entire surplus value, so it is a system with “simple reproduction”. The obvious disadvantage of Bortkiewicz's solution is the quantitative difference between the value and price sum, which occurs in every other case except in the very special case of the same organic composition in all three sectors or in the case of a surplus value of zero.

Bortkiewicz's proposed solution was in principle anticipated and published by the social democrat Wolfgang Mühlpfordt 12 years before Bortkiewicz, and thus before the publication of the third volume of Marx's “Capital”. Mühlpfordt's study was forgotten for almost a century. Bortkiewicz's algorithm was improved several times in the course of the debate and expanded by Francis Seton from a solution with three sectors to a solution with any number of sectors. This phase of the discussion came to an end with Seton's 1956 contribution. A similar alternative method was then developed by Sraffa , which AWT completely abandoned, instead made the material interweaving structure of "goods production by means of goods" as the basis and developed a price system from this, which determines the prices of goods independently of subjective benefit estimates . Sraffa renounces the AWT, even if he provides the mathematical basis for the final solution of the transformation problem from the point of view of equilibrium economics.

Whether the solution by Bortkiewicz solves the transformation problem is controversial. Friedrun Quaas, for example, believes that the Bortkiewicz method makes it possible to consistently derive production prices from labor values” .

The contribution by Carl Christian von Weizsäcker

Carl Christian von Weizsäcker investigated the problem (1962, 1971) assuming that the economy grows at a rate according to the so-called golden rule of accumulation . He comes to the conclusion that production prices can be represented as labor values. However, the production prices of an end product are not simply proportional to the sum of the labor values ​​of all products that are necessary for the manufacture of the end product. Rather, earlier labor values ​​are “paid interest” up to the point in time of the completion of the end product, whereby the rate of profit, which is equal to the rate of economic growth when the “golden rule” applies, serves as the interest rate.

Marxist responses to the "Sraffa" shock

However, since Sraffa's founding of modern Neoricardianism in 1960, it has been clear that Bortkiewicz's algorithm can ultimately be transformed into a system of equations in which labor values ​​only occur as redundant by-products. For the calculation of equilibrium prices, they do not have a formal priority status.

This "Sraffa shock" elicited various responses from the Marxist side:

  1. AWT was completely abandoned by (former) Marxists like Ian Steedman and instead the focus was placed on the theory of the physical surplus product and the class-based distribution of the surplus , which can be formulated without recourse to labor values . In 1978, Hans-Georg Sprotte was able to provide formal proof that even with full acceptance of the Sraffa algorithm, the AWT can be rehabilitated in the event of equilibrium growth, because in this case the surplus product is distributed so proportionally to all system components that its composition corresponds to the social Total product in miniature. The normalization of the surplus product to a “standard product” through the “ golden rule of accumulation ” solves the transformation problem, but means the abandonment of any crisis-theoretical application of the AWT. This growth-theoretical solution to the transformation problem was presented by Michio Morishima in 1973.
  2. By Michio Morishima the neoricardianische criticism was accepted, but rejected the fundamental rejection of AWT. After Morishima in fact the value-shaped exploitation thesis can be justified in so far as a positive rate of profit (and exploitation in the Marxian sense) must always correlate with the existence of a positive added value. The deviation of the price-based from the value-based surplus product is thus given an objective limit and consequently the link between profit and surplus value is guaranteed. Despite the transformation problem, profit can ultimately always be traced back to unpaid overtime . Morishima called this fact “Marx's fundamental theorem” (MFT). In the wake of Morishima's theses on “Marx's fundamental theorem”, a debate developed about whether under special conditions (“ joint production ”) positive surplus value quanta could correlate with negative profits and vice versa. The transformation problem would then to a certain extent be “taken to extremes”, because now positive values ​​after the transformation could result in negative price values ​​(and vice versa), which would obviously lead to completely nonsensical interpretations of this transformation procedure. This debate was triggered in 1975 by Ian Steedman's criticism of the MFT by means of a numerically presented mind game. Morishima met this criticism with a counter criticism of the examples Ian Steedmans and the formulation of " the General Marxian Fundamentaltheorems" (GMFT), which remains valid even in the case of joint production. Ultimately, in the course of this debate, each party considered its interpretation of the AWT to be confirmed. From Steedman's point of view , the MFT was refuted: Morishima's attempted solutions - which are based on a reinterpretation of “socially necessary working time” into “socially optimal ” values ​​by means of linear optimization - changed the content of the AWT in his view so that the subject area no longer corresponds to Marx's Labor theory of value is compatible. From Morishima's point of view , on the one hand, Steedman's examples were logically untenable or unrealistic and, on the other hand, could be integrated in the present form through a simple extension of the MFT to the GMFT (which takes into account joint production) and thus no fundamental problem for the AWT. The question of whether the GMFT can now refer to the Marxian texts to a very limited extent must undoubtedly be decided in favor of Steedman - Morishima could only refer to a passage from Marx's “Misery of Philosophy” - a text in support of the GMFT , which was written shortly before the "Communist Manifesto" (more precisely: from December 1846 to June 1847). On the question, however, whether the GMFT in itself is consistent, this debate on Marx exegesis irrelevant.
  3. By authors such as Paul Mattick of the explanatory power of the AWT has been modified so that the transformation problem is ultimately disappeared in his quantitative form. A more recent version of this interpretation of the transformation problem is represented by Michael Heinrich , who on the one hand accepts the simultaneous formalization of value theory (that ultimately price structures refer to price structures), but on the other hand regards value form analysis as the central core content of AWT. As a result, Sraffa prices can be traced back to abstract work ex post using value form analysis and the transformation problem can be reversed or resolved. By means of the Neoricardian models, such a mathematical return of all price components including the surplus product to labor values ​​is expressly possible, the labor values ​​are only formally superfluous for the determination of the equilibrium prices. A quantitative problem of value-price calculation no longer exists in Heinrich's “monetary value theory”, because a redistribution of labor values ​​(for calculating production prices in the “forward movement”) as in the current conversion models, according to Heinrich, sets a premonetary understanding of Marx's labor value theory ahead. If, however, the value is consistently thought of as a monetary phenomenon - according to the value form analysis - labor values ​​can only appear in the form of monetary production prices, because these are the monetary expressions of the deeper production relationships (which are reflected in value theory). All transformation algorithms come down to the implicit idea of ​​premonitarily existing value quanta, which outside of their appearance (namely the money form) already enter into quantitative causal relationships. Accordingly, Heinrich rejects the part of the debate started by Bortkiewicz because of the premonetary concept of value used in this discourse. Heinrich interprets the relationship between values ​​and prices epistemologically as a " conceptual development relationship " which expresses "the formal content of the exchange between capital and labor."
  4. In principle, the problem and the "not completed" Marxian method were accepted, better proposed solutions that, unlike the neo-Cardian method, support the AWT. In particular, the New Solution of the Transformation Problem by the authors Dumenil , Foley and Lipietz , received and discussed by v. a. in the Anglo-American region, has set an example here. As part of the New Solution , the transformation is limited to the net product and the value of the workforce is no longer interpreted as real wages , but as money wages . Thus, the value of the commodity labor is not determined by a given bundle of consumer goods, but as money wage income, which represents a general claim to a part of the total output. The share of wages in national income is therefore equated by assumption in the value system and in the price system. Interestingly, this solution was anticipated by Joan Robinson back in the 1960s , but went largely unnoticed.
  5. The two Israeli mathematicians Emmanuel Farjoun and Moshe Machover circumvent the transformation problem in its “classical” form while maintaining the AWT by rejecting the construct of a general rate of profit and working with a stochastic approach as an alternative . Farjoun and Machover point out that capitalist competition creates mechanisms that both push in the direction of an average rate of profit and include the "countermovements" to the destruction of the general rate of profit. It is therefore more realistic to assume that a “profit rate spectrum” exists that can be captured theoretically (with Farjoun / Machover using a curve of the gamma distribution ) and not deterministically using linear algebra . The center of fluctuation in the commodity prices determined by Farjoun and Machover is very close to the pure labor values ​​of the first volume of capital . A number of empirical studies support this view and Fritz Helmedag also pointed out the consistency and efficiency of the “pure” AWT in his study “Goods production by means of work”, although Helmedag does not take a probabilistic approach. Farjoun's and Machover's study generated a far greater response in Anglo-American literature than in German-speaking countries, where their approach remained almost unknown (their main work "Laws of Chaos, A Probabilistic Approach to Political Economy" has never been translated into German)
  6. A more recently, z. An approach from the Anglo-American region, for example, represented by Andrew Kliman (2007) or Alan Freeman (1996) represents the " Temporal Single System Interpretation " (TSSI) of the transformation problem. This approach generally rejects the view that linear systems of equations are an adequate methodological procedure for provide a meaningful representation of the relationship between values ​​and prices. In contrast to this, these authors do not interpret the AWT "simultaneously", i.e. calculating inputs and outputs of the production process at the same time, but "temporally". The reproduction process of capital is understood here in the form of a causal-time, which in turn eliminates the requirement that input and output prices must be identical. Second, the term “single system” indicates that these authors do not understand value and price level as two strictly separate functional areas (“dualistic interpretation”), but as inseparable. Here, the values ​​of the input goods are understood as monetary evaluations at a certain causal point in the movement of capital, whereby this evaluation is not carried out in two ways. As monetary values, the values ​​of the inputs are de facto price values, because inputs are firmly fixed through monetary investments and no longer quantitatively re-evaluated prices for acquired means of production in the course of the further exploitation movement. Here the TSSI follows on from the New Solution , whereby in the New Solution only money wages are interpreted in monetary terms, with the result that in this model means of production are obviously subject to the natural exchange mode and wages are paid with money. The TSSI, on the other hand, understands constant and variable capital in monetary terms, as investment quantities paid in money (and thus quantitatively invariant in the further utilization process). In the TSSI, output prices cannot have simultaneous repercussions on input prices of an already completed production period, but can only determine input prices in the next (causally-temporally following) exploitation movement. With a causal-temporal interpretation of Marx's transformation process, the classic problems discussed since Bortkiewicz are resolved in the opinion of these authors. Accordingly, they refer to Bortkiewicz's close connection to the general equilibrium theory of neoclassics and devote a lot of space to the criticism of this neoclassical interpretation of Marx.


  1. ^ Ian Steedman: Value, Price and Profit. New Left Review I / 90, March-April 1975.
  2. cf. on this Ronald L. Meek : Some remarks on the transformation problem. in: Friedrich Eberle, (Ed.): Aspects of Marx's theory 1. On the methodological significance of the third volume of 'Capital'. Frankfurt 1973. pp. 255–274 (from the English by Joska Fischer and Jürgen Ritsert ; from: Economics and Ideology and other essays. London 1967)
  3. ^ PA Samuelson, 1974, p. 239
  4. Eberhard Feess-Dörr: The redundancy of the added value theory. A contribution to the controversy between Marxists and Neoricardians. Marburg 1989.
  5. ^ Paul M. Sweezy, 1971, p. 148.
  6. ^ Weizsäcker, Carl Christian von (2010): A New Technical Progress Function (1962). German Economic Review 11/3 (first publication of an article written in 1962); CC Weizsäcker, PA Samuelson: A new labor theory of value for rational planning through use of the bourgeois profit rate. In: Proceedings of the National Academy of Sciences . Volume 68, Number 6, June 1971, pp. 1192-1194, PMID 16591926 , PMC 389151 (free full text).
  7. Michael Heinrich (1999), p. 281 ff.
  8. Fröhlich, Nils: The Review of Classical Price Theories Using Input-Output Tables, Economy and Statistics 5/2010, pp. 503–508. "All in all, against the background of the German input-output tables from 2000 and 2004, there is a lot to suggest that real prices are actually regulated by the classic law of value - ..."
  9. To mark the 25th anniversary of the first publication of “laws of chaos”, there is even a conference at Kingston University, UK. Sh.
  10. Emmanuel Farjoun and Moshe Machover, Laws of Chaos; A Probabilistic Approach to Political Economy , London: Verso, 1983. Free verso books .


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