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Line 1: {{Short description\|Pricing problem in Marxism}} {{About\|the transformation problem in economics\|the transformation problem in group theory\|Conjugacy problem}} {{Multiple issues\| {{more footnotes needed\|date=October 2014}} {{original research\|date=July 2020}} }} {{Marxian economics\|expanded=Topics}} In 20th-century discussions of [[Karl Marx]]'s [[Marxian economics\|economics]], the '''transformation problem''' is the problem of finding a general rule by which to transform the "values" of commodities (based on their socially necessary labour content, according to his [[labour theory of value]]) into the "competitive prices" of the marketplace. This problem was first introduced by Marxist economist [[Conrad Schmidt (economist)\|Conrad Schmidt]]<ref>{{Cite book \|last=Böhm-Bawerk \|first=Eugen \|title=Karl Marx and the Close of his System \|year=1896 \|publisher=CreateSpace Independent Publishing Platform \|isbn=978-1466347687 \|language=en}}</ref> and later dealt with by Marx in chapter 9 of the draft of [[Capital, Volume III\|volume 3 of ''Capital'']]~~, where he also sketched a solution~~. The essential difficulty was this: given that Marx derived profit, in the form of [[surplus value]], from direct labour inputs, and that the ratio of direct labour input to capital input varied widely between commodities, how could he reconcile this with a tendency toward an average rate of profit on all capital invested among industries, if such a tendency (as predicted by Marx and Ricardo) exists? == Marx's theory == Marx defines [[Exchange value\|value]] as the number of hours of labor socially necessary to produce a commodity. This includes two elements: First, it includes the hours that a worker of normal skill and dedication would take to produce a commodity under average conditions and with the usual equipment (Marx terms this "living labor"). Second, it includes the labor embodied in raw materials, tools, and machinery used up or worn away during its production (which Marx terms "dead labor"). In capitalism, workers spend a portion of their working day reproducing the value of their means of subsistence, represented as wages (necessary labor), and a portion of their day producing value above and beyond that, referred to as [[surplus value]], which goes to the capitalist (surplus labor). Since, according to Marx, the source of capitalist profit is this [[surplus labor]] of the workers, and since in this theory only new, living labor produces ~~profit~~value, it would appear logical that enterprises with a low [[Organic composition of capital\|organic composition]] (a higher proportion of capital spent on living labor) would have a higher rate of profit than would enterprises with a high organic composition (a higher proportion of capital spent on raw materials and means of production). However, in models of classical perfect competition, higher rates of profit are not generally found in enterprises with a low organic composition, and low profit rates are not generally found in enterprises with a high organic composition.{{citation needed\|date=May 2024}} Instead, there is a tendency toward [[Surplus value#Equalization of rates of surplus value\|equalization of the rate of profit]] in industries of different organic compositions. That is, in such models with no barriers to entry, capitalists are free to disinvest or invest in any industry, a tendency exists towards the formation of a general rate of profits, constant across all industries. Marx outlined the transformation problem as a theoretical solution to this discrepancy. The tendency of the rate of profit toward equalization means that, in this theory, there is no simple translation from value to money—e.g., ''1 hour of value equals 20 dollars''—that is the same across every sector of the economy. While such a simple translation may hold approximately true in general, Marx postulated that there is an economy-wide, systematic deviation according to the organic compositions of the different industries, such that ''1 hour of value equals 20 dollars times T'', where ''T'' represents a transformation factor that varies according to the organic composition of the industry in consideration. Line 95 ⟶ 96: for all <math>i</math>. This necessarily uniform ratio <math>{s_i \over v_i} = \sigma</math> is called by Marx the [[rate of ~~surplus value~~exploitation]], and it allows to re-write Marx's value equations as: :<math>p_i = c_i + v_i (1 + \sigma) = l_A a_i + l_W l_i (1 + \sigma)</math> Line 165 ⟶ 166: Tables 1 and 2 parallel the tables in which Marx elaborated his numerical example.<ref>[http://www.marxists.org/archive/marx/works/1894-c3/ch09.htm Capital, III Chapter 9]</ref> === Marx's supposed error and its correction === Later scholars argued that Marx's formulas for competitive prices were mistaken. Line 199 ⟶ 200: There are several schools of thought among those who see themselves as upholding or furthering Marx on the question of transformation from values to prices, or modifying his theory in ways to make it more consistent. According to the [[temporal single-system interpretation]] of ''Capital'' advanced by Alan Freeman, Andrew Kliman, and others, Marx's writings on the subject ~~can~~are bemost robustly interpreted in such a way as to remove any supposed inconsistencies (.{{sfn\|Choonara \|2007).}} Modern traditional Marxists argue that not only does the labour theory of value hold up today, but also that Marx's understanding of the transformation problem was in the main correct. Andrew Kliman claimed using the TSSI framework: "Simple reproduction and uniform profitability do require that supplies equal demands, but they can be equal even if the input and output prices of Period 1 are unequal. Since the outputs of one period are the inputs of the next, what is needed in order for supplies to equal demands is that the output prices of Period 1 equal the input prices of Period 2. But they are always equal; the end of one period is the start of the next, so the output prices of one period necessarily equal the input prices of the next period. Once this is recognized, Bortkiewicz’s proofs immediately fail, as was first demonstrated in Kliman and McGIone (1988)".<ref name=JG>Joseph Green (2010): [http://www.communistvoice.org/45cTransformation1.html On the non-naturalness of value: A defense of Marx and Engels on the transformation problem (part one)]</ref> Political-economic readings of ''Capital'', such as [[Harry Cleaver]]'s ''[[Reading Capital Politically]]'' redefine exploitation as a direct control of worked time, unrelated as such to distribution. These readings are usually associated with the [[Autonomist Marxism\|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 consider the transformation problem unimportant because they see all social arrangements in capitalism (in particular, profit and distribution) as politically determined contests between classes. In the probabilistic interpretation of Marx advanced by Emmanuel Farjoun and Moshe Machover in ''Laws of Chaos'' (see references), they "''dis''solve" the transformation problem by reconceptualising the relevant quantities as random variables. In particular, they consider profit rates to reach an equilibrium ''distribution''. 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 (Economist)\|Anwar Shaikh]]~~, Fred Moseley, Alan Freeman~~, [[Makoto Itoh]], Gerard Dumenil and Dominique Levy, and Duncan Foley) who hold 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. 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 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 they hold that the practical details of their workings are more complicated than Marx thought. In particular, some (e.g., [[Anwar Shaikh (Economist)\|Anwar Shaikh]]) 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. Using input-output data and empirical proxies for labour-values, Shaikh and Ochoa have provided some statistical evidence to show that, although no incontestable ''logical'' deduction may be 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. == Critics of the theory == === Marxian ~~reply~~and ~~to non-Marxian~~Sraffian critiques ===▼ ~~Many~~Some mathematical economists assert that a set of functions in which Marx's equalities hold does ''not'' generally exist at the individual enterprise or aggregate level, so that chapter 9's transformation problem has no general solution, outside two very special cases. This was first pointed out by, among others, ~~[[Eugen von Böhm-Bawerk\|Böhm-Bawerk]] (1896) and~~ [[Ladislaus Bortkiewicz\|Bortkiewicz]] (1906). In the second half of the 20th century, [[Wassily Leontief\|Leontief]]’s and [[Piero Sraffa\|Sraffa]]’s work on linear production models provided a framework within which to ~~prove~~argue this result in a ~~simple and~~ general way. Although he never actually mentioned the transformation problem, Sraffa’s (1960) chapter 6 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 <math>P_i</math> of the <math>i^{th}</math> produced good can be expressed as Line 224 ⟶ 222: A standard reference, with an extensive survey of the entire literature prior to 1971 and a comprehensive bibliography, is [[Paul Samuelson\|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. Proponents of the temporal single system interpretation such as Moseley (1999), 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, ~~question~~reject ~~whether~~the principles of the mathematical proof that Marx's transformation problem has no general solution. 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 a controversial issue today~~.▼ 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]] held 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. 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. The lack of any function to transform Marx's "values" to competitive prices has important implications for Marx's theory of labour [[Exploitation of labour\|exploitation]] and [[capital accumulation\|economic dynamics]]—namely that, some people argue with [[Okishio's theorem]], that there is no [[tendency of the rate of profit to fall]]. This means that it is not preordained 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. ▲Proponents of the temporal single system interpretation such as Moseley (1999), 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, question whether the mathematical proof that Marx's transformation problem has no general solution. 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 a controversial issue today. Simon Mohun and Roberto Veneziani explain that if the LTV is taken as a macroeconomic theory, then total revenue will equal total value. However, this will mean that total profit will not be proportional to total surplus-value, meaning that surplus value cannot be the explanation for the origins of profit. However, if there is a proportionality between surplus value and profit, then the LTV fails as a macroeconomic theory.<ref>Mohun, Simon, and Roberto Veneziani. "Value, price, and exploitation: The logic of the transformation problem." Journal of Economic Surveys 31, no. 5 (2017): 1387-1420.</ref> === Non-Marxian critiques === Mainstream scholars such as [[Paul 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 applies, Marx's reasoning can be turned upside down through an inverse transformation process; Samuelson argues that Marx's inference that <blockquote>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 3, chapter 2)</blockquote> could with equal cogency be "transformed" into: Line 241 ⟶ 233: <blockquote>Surplus value is therefore the [Marxist] disguise of profit which must be removed before the real nature of profit can be discovered.<ref>Samuelson (1971), p. 417</ref></blockquote> Samuelson not only dismissed the labour theory of value because of the transformation problem, but provided himself, in cooperation with economists like [[Carl Christian von Weizsäcker]], solutions. Von Weizsäcker (1962),<ref name=CW>Weizsäcker, Carl Christian von (2010): A New Technical Progress Function (1962). German Economic Review 11/3 (first publication of an article written in 1962)</ref> along with Samuelson (1971),<ref name=WS>Weizsäcker Carl Christian von, and [[Paul A. Samuelson]] (1971): A new labor theory of value for rational planning through use of the bourgeois profit rate. Proceedings of the National ~~Acadademy~~Academy of Sciences U S A. [https://www.ncbi.nlm.nih.gov/pmc/articles/PMC389151/ download of facsimile]</ref> analysed the problem under the assumption that the economy grows at a constant rate following the [[Golden Rule savings rate\|Golden Rule of Accumulation]]. Weizsäcker concludes:▼ To clarify this point, it may be noticed that the special cases in question are also precisely those where [[John Bates Clark\|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, a hypothesis regarded as disproved by all sides during the [[Cambridge capital controversy]]. One would thus have a "pure" state of capitalist society where Marx's [[exploitation theory]] and its main supposed confutation were both true. 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. ▲Samuelson not only dismissed the labour theory of value because of the transformation problem, but provided himself, in cooperation with economists like [[Carl Christian von Weizsäcker]], solutions. Von Weizsäcker (1962),<ref name=CW>Weizsäcker, Carl Christian von (2010): A New Technical Progress Function (1962). German Economic Review 11/3 (first publication of an article written in 1962)</ref> along with Samuelson (1971),<ref name=WS>Weizsäcker Carl Christian von, and [[Paul A. Samuelson]] (1971): A new labor theory of value for rational planning through use of the bourgeois profit rate. Proceedings of the National Acadademy of Sciences U S A. [https://www.ncbi.nlm.nih.gov/pmc/articles/PMC389151/ download of facsimile]</ref> analysed the problem under the assumption that the economy grows at a constant rate following the [[Golden Rule savings rate\|Golden Rule of Accumulation]]. Weizsäcker concludes: ~~<blockquote>The price of the commodity today is equal to the sum of the 'present' values of the different labour inputs.<ref>Weizsäcker (2010 [1962]), p. 262</ref></blockquote>~~ ▲=== Marxian reply to non-Marxian critiques === The Marxian reply to this mainstream view is as follows. The attempt to discard the theoretical relevance of the necessary preconditions of Marx's value analysis in volume 1 of ''Capital'' through a [[reductio ad absurdum]] is superficial. By first identifying that the preconditions necessary for J. B. Clark's old model of ''aggregate'' marginal productivity to hold true are the same as those necessary for Marxian values to conform to relative prices, we are then supposed to conclude that the foundation of Marx's analysis ''as based in these preconditions'' is faulty ''because'' Clark's model had been proven wrong in the Cambridge capital controversy. The superficiality stems from the fact that those who support this reduction forget that the Cambridge capital controversy called the entire concept of marginal productivity into question by attacking not Clark's special case assumptions but the notion that physical capital can be aggregated. Marx simply does not run into this problem because his analysis does not rely on an aggregation of physical quantities that receive a return based on their contribution as "factors" of production. The fact that marginal productivity in its aggregate form is "a hypothesis regarded as disproved by all sides during the Cambridge capital controversy" has nothing to do with the validity of the special cases of Marx, and thus we ''would not'' "have a "pure" state of capitalist society where Marx's [[exploitation theory]] and its main supposed confutation (Clark) were ''both'' true", as is concluded from this view,'' because'' the "correctness" or "incorrectness" of Clark's aggregate marginal productivity scheme in this case flows not from special case assumptions but from the fact that he is aggregating physical units of capital; i.e., Clark's argument would still not hold true ''even with'' the assumed special cases. To further clarify this point, consider the following. First, 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 that go beyond facts and logic. By nature, the concept of economic value is not a scientifically provable concept but an assumption. Marx himself explicitly ridiculed the idea that he should be required to "prove his concept of value". Second, 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 that it made possible (see also [[law of value]]). What mattered was the ''application'' of the concept. Third, 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 observable evidence. And that concept of value can be compared with rival concepts and the rival theories they make possible in order to establish which has greater explanatory or predictive capacity. Fourth, modern [[philosophy of science]] rejects Popper's falsification theory as an adequate portrayal of science. Scientific statements are not necessarily ''falsifiable'' statements but ''fallible'' statements (i.e., they could be wrong) that, in principle, can be tested against observables, even if we do not yet know technically how to do this. Scientists do not aim mainly to falsify theories, but to confirm them in order to provide usable knowledge.<ref>{{Cite book\|last=Sheehan\|first=Helena\|title=Marxism and the Philosophy of Science: A Critical History\|publisher=Verso\|year=2018}}</ref> <blockquote>The price of the commodity today is equal to the sum of the 'present' values of the different labour inputs.<ref>Weizsäcker (2010 [1962]), p. 262</ref></blockquote>Even during the 19th century, [[Austrian school of economics\|Austrian economist]] [[Eugen von Böhm-Bawerk]] criticizes Marx's solution as being inconsistent : while in the first chapter of the first volume of The Capital Karl Marx explained that the value of any commodity was generally reflected by the quantity of labor required, inequality being only a temporary exception, this therefore means that the level of value generated is completely independent of the quantity of capital of a company, in other words, the [[organic composition of capital]] (i.e. the ratio between the quantity of capital and the quantity of labor) of a company has no impact on the profits generated.<ref>{{Cite book \|last=Böhm-Bawerk \|first=Eugen \|title=Karl Marx and the Close of his System \|year=1896 \|isbn=978-1466347687 \|pages=13 \|publisher=CreateSpace Independent Publishing Platform \|language=en \|quote=According [to Karl Marx], given an equal rate of surplus value, every branch of production must show a different, a special rate of profit, on the condition certainly, which Marx has hitherto always assumed, that commodities exchange with each other 'according to their values', or in proportion to the work embodied in them.}}</ref> However when faced to the transformation problem, Karl Marx is forced to reconsider his thesis, thus he explains in the third volume of Capital that after production, capitalists will reallocate their capital towards companies having made the highest rates of surplus value until the rate of surplus value stabilizes for all companies in a sector of production (since capital is not a source of value and therefore of profit for Marx), thus, the prices of goods will go from 'induced' by the value of labor to ''price of production'' (the sum of wages and annual profits), "The value and price of the commodity coincide only accidentally and exceptionally." However, Böhm-Bawerk pinpoints the contradiction formulated with the relation between the value and the price of the good in the first volume, thus, the Marxist theory appears contradictory and the [[labor theory of value]] illogical.<ref>{{Cite book \|last=Böhm-Bawerk \|first=Eugen \|title=Karl Marx and the Close of his System \|year=1896 \|isbn=978-1466347687 \|pages=19 \|publisher=CreateSpace Independent Publishing Platform \|language=en \|quote=The value [of labour] was declared to be 'the common factor which appears in the exchange relation of commodities' (i. 13). We were told, in the form and with the emphasis of a stringent syllogistic conclusion, allowing of no exception, that to set down two commodities as equivalents in exchange implied that 'a common factor of the same magnitude' existed in both, to which each of the two 'must be reducible' (i. 11). (...) And now in the third volume (...) that individual commodities do and must exchange with each other in a proportion different from that of the labour incorporated in them, and this not accidentally and temporarily, but of necessity and permanently. I cannot help myself; I see here no explanation and reconciliation of a contradiction, but the bare contradiction itself. Marx's third volume contradicts the first. The theory of the average rate of profit and of the prices of production cannot be reconciled with the theory of value. This is the impression which must, I believe, be received by every logical thinker. And it seems to have been very generally accepted. Loria, in his lively and picturesque style, states that he feels himself forced to the 'harsh but just judgment' that Marx 'instead of a solution has presented a mystification.'}}</ref> 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 the exploitation of labour. 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 separate issue. == See also == * [[Capital accumulation]] * [[Critique of political economy]] * [[Labour theory of value]] * [[Law of value]] * [[Prices of production]] * [[Return of capital]] * [[Socially necessary labour time]] * [[Surplus value]] * [[Temporal single-system interpretation]] == Notes == Line 287 ⟶ 258: * {{cite journal \| url=https://commons.wikimedia.org/wiki/File:Bortkiewicz.1907c.pdf \| author=Ladislaus von Bortkiewicz \| title=Zur Berichtigung der grundlegenden theoretischen Konstruktion von Marx im dritten Band des 'Kapital' \| journal=Jahrbücher für Nationalökonomie und Statistik \|series=III Folge \| volume=34 \| pages=319–335 \| year=1907 \| language=DE }} * {{cite book \| author=Ladislaus von Bortkiewicz \| chapter=On the Correction of Marx's Fundamental Theoretical Construction in the Third Volume of ``Capital \| pages=197–221 \| editor=Paul M. Sweezy \| title=Karl Marx and the Close of his System \| location=New York \| publisher=Augustus M. Kelley \| year=1949 \| chapter-url=https://commons.wikimedia.org/w/index.php?title=File:Bortkiewicz.1949.pdf }} * ~~Choonara,~~{{cite web\|last=Choonara\|first=J. (\|year=2007~~) [~~\|url=http://www.isj.org.uk/index.php4?id=353&issue=115 "\|title=Marx's "transformation" made easy"]}} * Alan Freeman: ''Price, value and profit - a continuous, general treatment.'' In: Alan Freeman, Guglielmo Carchedi (editors): ''Marx and non-equilibrium economics.'' Edward Elgar. Cheltenham, UK, Brookfield, US 1996. * Meek, R. (1956) 'Some Notes on the Transformation Problem' ''Economic Journal'' '''66''' (March) 94-107. Line 310 ⟶ 281: {{DEFAULTSORT:Transformation Problem}} [[Category:Marxist theory]] [[Category:History of economic thought]] [[Category:Marxian economics]]