Editing Transformation problem (section)

== Critics of the theory ==
===Marxian and Sraffian critiques===
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, [[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 argue this result in a 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

:<math>P_i = \sum_{n=0}^\infty l_{in} w {(1+r)^n}</math>,

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

:<math>E_i = \sum_{n=0}^\infty l_{in}</math>,

it follows from Sraffa’s result that there is generally no function from <math>E_i</math> to <math>P_i</math>, 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 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&ndash;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, reject 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.

=== 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:

<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 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:

<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>