This problem was first introduced by Marxist economist Conrad Schmidt[1] and later dealt with by Marx in chapter 9 of the draft of volume 3 of Capital.
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 value, it would appear logical that enterprises with a low 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.
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.
Assume a hunters’ economy with free land, no slavery, and no significant current production of tools, in which beavers
Marx defined the "value" of a commodity as the total amount of socially necessary labour embodied in its production.
Due to the influence of Marx's particular definition of value on the transformation problem, he is quoted at length where he argues as follows: Let us take two commodities, e.g., corn and iron.
As labour produces in this sense more than its own value, the direct-labour input is called variable capital and denoted as
By contrast, the value of other inputs—in our example, the indirect (or "dead") past labour embodied in the used-up arrows—is transmitted to the product as it stands, without additions.
Table 1 shows how the total amount of surplus value of the system, shown in the last row, is determined.
Table 2 illustrates how Marx thought this total would be redistributed between the two industries, as "profit" at a uniform return rate, r, over constant capital.
Marx noted this but thought that it was not significant, stating in chapter 9 of volume 3 of Capital that "Our present analysis does not necessitate a closer examination of this point."
The simultaneous linear equations method of computing competitive (relative) prices in an equilibrium economy is today very well known.
Ernest Mandel, defending Marx, explains this discrepancy in term of the time frame of production rather than as a logical error; i.e., in this simplified model, capital goods are purchased at a labour value price, but final products are sold under prices that reflect redistributed surplus value.
In particular, the "law of value" would have prevailed in pre-capitalist exchange economies, from Babylon to the 15th century, while the "transformed" prices would have materialized under capitalism: see Engels's quotation by Morishima and Catephores (1975), p. 310.
These authors argued that, whatever one might say of his interpretation of capitalism, Marx's "value" theory retains its usefulness as a tool to interpret pre-capitalist societies, because, they maintained, in pre-capitalist exchange economies there were no "prices of production" with a uniform rate of return (or "profit") on capital.
It hence follows that Marx's transformation must have had a historical dimension, given by the actual transition to capitalist production (and no more Marxian "values") at the beginning of the modern era.
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.
[5] 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.
[6] In the probabilistic interpretation of Marx advanced by Emmanuel Farjoun and Moshe Machover in Laws of Chaos (see references), they "dissolve" the transformation problem by reconceptualising the relevant quantities as random variables.
Finally, there are Marxist scholars (e.g., Anwar Shaikh, 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.
In the second half of the 20th century, Leontief’s and Sraffa’s work on linear production models provided a framework within which to argue this result in a general way.
A standard reference, with an extensive survey of the entire literature prior to 1971 and a comprehensive bibliography, is 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, reject the principles of the mathematical proof that Marx's transformation problem has no general solution.
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 Profit is therefore the [bourgeois] disguise of surplus value which must be removed before the real nature of surplus value can be discovered.
[7]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),[8] along with Samuelson (1971),[9] analysed the problem under the assumption that the economy grows at a constant rate following the Golden Rule of Accumulation.
Weizsäcker concludes: The price of the commodity today is equal to the sum of the 'present' values of the different labour inputs.