Editing Transformation problem (section)

===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.