Transformation problem
전형 문제
The problem of finding a general rule that transforms the values of commodities given by the labour theory of value into the competitive prices at which profit rates are equal, that is, prices of production, and the debate of more than a century over it. In Chapter 9 of Volume III of Capital Marx obtained prices of production by redistributing total surplus-value at the average rate of profit, but still reckoned the inputs in the cost-price at their values. In 1896 Böhm-Bawerk charged that Volumes I and III contradict each other; when Bortkiewicz "corrected" Marx with simultaneous equations in 1906–07, the two aggregate equalities no longer held together. After Sraffa, critics argued that values are redundant for calculating prices, and Marxists replied with the New Interpretation, iterative procedures and the temporal single-system interpretation (TSSI).
In depth
What is the problem?
In Marx's table of five branches, described under price of production, each price of production was found by adding the average profit of 22 percent to the cost-price. But much of the cost-price consists of raw materials and machines bought from other branches and of the means of subsistence that workers buy, and these too sell on the market at prices of production, not at values. The cost-price should then also be reckoned at prices of production, and that changes the average rate of profit and every price again. Marx knew this. After noting that "there is always the possibility of an error if the cost-price of a commodity in any particular sphere is identified with the value of the means of production consumed by it", he wrote: "Our present analysis does not necessitate a closer examination of this point."
If inputs are transformed into prices of production as well, do Marx's two equalities, total prices equal total values and total profit equals total surplus-value, still hold together? And if they do not, what standing does the theory of exploitation, the claim that surplus-value is the source of profit, have as a theory of prices and profits? That is the transformation problem.
What did Böhm-Bawerk criticise?
Engels put the riddle up for public discussion in his prefaces to Volumes II and III, and objections poured in once Volume III appeared. The most influential was Karl Marx and the Close of His System (1896) by the Austrian School economist Eugen von Böhm-Bawerk. Marx, he argued, said in Volume I that the value of commodities is determined by labour and in Volume III that commodities sell not at their values but at prices of production, so his law of value contradicts itself. The book is considered the "classical" critique of Capital, and most later critiques repeated its arguments.
Rudolf Hilferding answered for the Marxists with Böhm-Bawerk's Criticism of Marx, treating value theory not as a theory that predicts individual prices directly but as an account of how social labour is allocated. This controversy did not yet deal with the transformation problem in the narrow, mathematically formalised sense.
What did Bortkiewicz correct?
In 1906–07 the statistician Ladislaus von Bortkiewicz set up simultaneous equations for three departments producing means of production, workers' consumer goods and luxuries, reckoning both inputs and outputs at prices of production. Three equations with four unknowns (three price coefficients and the rate of profit) need one more condition, and he treated gold as the luxury good and fixed its price coefficient at one. Total profit then equals total surplus-value, but total price as a rule differs from total value. He showed that Marx's own variables suffice to calculate the rate of profit and relative prices, and at the same time that Marx's two equalities cannot both hold.
This "correction" went almost unnoticed until Paul Sweezy presented it in The Theory of Capitalist Development in 1942. Sweezy favoured Bortkiewicz's solution and in 1949 brought out Böhm-Bawerk's and Hilferding's texts together with Bortkiewicz's article in one volume. In 1956 Francis Seton generalised the solution to any number of sectors, which closed this phase of the discussion.
What was the redundancy critique after Sraffa?
Piero Sraffa's Production of Commodities by Means of Commodities (1960) offered a system that calculates prices of production and the rate of profit from physical input coefficients and the real wage alone, without labour values. In Marx after Sraffa (1977) Ian Steedman concluded that the value calculation is a redundant detour for explaining prices and profits and abandoned the labour theory of value. Paul Samuelson in 1971 described the procedure as erasing the value system and writing in the price system, and quipped that Marx's reasoning could equally be turned upside down, from prices to values.
Michio Morishima accepted the critique but defended the theory of exploitation. By what he named the Fundamental Marxian Theorem, first proved by Nobuo Okishio in 1955, profit is positive if and only if surplus-value is positive: however far individual prices deviate from values, profit still rests on unpaid surplus labour.
How did Marxists respond?
The answers diverged. In the early 1980s Gérard Duménil, Duncan Foley and Alain Lipietz put forward the "New Interpretation" (New Solution): if the transformation is confined to the year's net product and the value of labour-power is defined not as a given bundle of consumer goods but as the money wage, a claim to a share of total output, then the price total of the net product equals the new value created, and total surplus-value equals total profit. Anwar Shaikh, borrowing an idea from Morishima, showed by iteration that repeating Marx's procedure makes prices converge on prices of production. Shaikh, Makoto Itoh, Duménil and Dominique Lévy, and Foley hold that there is no incontestable logical procedure for deriving prices from values but that this is not fatal to Marx's system as a whole.
Another current rejects the problem as posed. The temporal single-system interpretation (TSSI), put forward from the early 1980s by Alan Freeman, Andrew Kliman, Guglielmo Carchedi and others, charges that the Bortkiewiczian reading determines input and output prices at the same moment (simultaneously) and treats values and prices as two independent systems. Read in temporal sequence and as one system in which values and prices determine each other, Marx's theory, they argue, is free of contradiction. Kliman explains: "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." Once this is recognised, Bortkiewicz's proofs fail, as Kliman and McGlone first argued in 1988. Critics such as David Laibman reply that the TSSI's proponents "assert that Marx's formulations ... are literally and completely correct", and many Marxist economists do not accept its conclusions.
Paul Mattick and, later, Michael Heinrich took value-form analysis as the core of value theory and so in effect dissolved the quantitative problem; Farjoun and Machover dropped the general rate of profit for a probabilistic approach; Ernest Mandel and others held that the problem arises only because simultaneous methods cannot capture the causal sequence in time of production and value formation.
Why has the debate lasted so long?
Because the central claim of Marxian economics is at stake: does the theory that profit comes from workers' unpaid labour explain the actual movement of prices and profits, or is it a moral claim that can be left out of the calculation? The same clash between simultaneous and temporal methods recurs in the debate over Okishio's theorem, widely taken to refute the law of the tendency of the rate of profit to fall.