The New Palgrave Dictionary of Economics

2018 Edition
| Editors: Macmillan Publishers Ltd

Value and Price

  • Meghnad Desai
Reference work entry


The problem of the relationship between value and price – the so called Transformation Problem – is a central issue in Marxian economics. In one sense it can be posed as a technical or mathematical problem of deriving a set of prices from a given set of value equations. But if it were only a technical problem then it should have a definite answer – either a solution exists or it does not. It is surprising therefore that this problem has continued to attract succeeding generations of economists since the date of publication of volume 3 of Capital in 1894 (Marx 1894).

The problem of the relationship between value and price – the so called Transformation Problem – is a central issue in Marxian economics. In one sense it can be posed as a technical or mathematical problem of deriving a set of prices from a given set of value equations. But if it were only a technical problem then it should have a definite answer – either a solution exists or it does not. It is surprising therefore that this problem has continued to attract succeeding generations of economists since the date of publication of volume 3 of Capital in 1894 (Marx 1894).

The debate shows no signs of abating and seems a rare example of a problem which continues to invite new solutions or versions in new mathematical language of the old solution. There can rarely have been a question in economic theory which has been solved so many times in so many different mathematical languages but yet not resolved finally. This continuing fascination of the Transformation Problem leads one to suspect that there is more than a technical issue at stake.

The locus classicus of the debate is chapter IX of Capital Vol. 3 (3/IX), which was published posthumously by Engels from notes left by Marx. There is evidence however that the material contained in this volume was written some time in the 1860s before the publication of Capital Vol. 1 (Marx 1867). This is of more than biographical interest in the debate. In Vol. 1, Marx developed his theory on the explicit assumption that values and prices were proportional to each other. This was done in awareness of two qualifying conditions; first that this was a special case and generally value and prices were related systematically but not proportionally, but second that values and value relations were unobservable, latent or structural whereas prices were observable, actual and phenomenal. The hidden nature of value relations – commodity fetishism – is crucial to Marx’s argument and hence it would have been totally uncharacteristic of Marx’s approach not to have foreseen that values and prices diverge from each other.

This divergence of prices from values emerged as a central result of 3/IX and was seized upon by Böhm-Bawerk in his Karl Marx and the Close of His System (1896; Sweezy 1949) as a basic deficiency and disproof of Marx’s theory of profits. He took it to be a complication that may have arisen in Marx’s work after he had written the first volume and an impression was conveyed that the price value divergence, being contrary to the proportionality assumed in Vol. 1, invalidated the conclusions in that volume.

If Böhm-Bawerk was able to gain and convey this impression it was because Marx’s attempt at solving the Transformation Problem looks unfinished. Having derived a numerical solution for prices from a set of value equations, as we will see below, Marx confronts the divergence as a puzzle and then spends some pages tacking around the problem but in no way presenting it as a systematic outcome. Thus it could be thought from reading 3/IX that the Transformation Problem was left unsolved.

The Problem

Marx’s theory of profit was that profits were the money form of surplus value produced by labour during the production process. The conversion of surplus value into profits was accomplished not at the level of the firm but of the whole economy. This conversion had to be effected in the context of a contractual purchase of labour by employers (i.e. no extraeconomic coercion) and secondly, the rate of profit had to be equal in all activities. The first consideration meant that the wage rate – the exchange value of the commodity sold by the labourer and bought by the employer – was determined on the same principles as any other commodity. Thus the existence of surplus value had to be reconciled with an economic determination of the exchange value of the commodity labour power.

To drive a wedge between the product of labour and its price, Marx used the accepted distinction between use value and exchange value of a commodity. The commodity in question, labour power, is the labourer’s potential for production. The use-value of labour power to the purchaser of the commodity – the capitalist employer – was measured in terms of the total labour time contracted to be spent by the labourer in production – the length of the working day in hours. The exchange value of labour power, like that of any other commodity, was the amount of labour time required for its reproduction, measured by the labour time equivalent of the basket of wage goods purchasable by the given wage. Having thus obtained two commensurable measures of the use value and the exchange value of labour power, the wedge between them was identified as surplus value, produced by the labourer but retained by the purchaser of labour power, the capitalist employer.

Now the total value of a commodity comprised the value contained in the materials used up in the production process – raw materials and energy used as well as the wear and tear of the fixed means of production – which Marx labelled constant capital (c) and the total value contributed by labourers. The latter consists of the exchange value of the wage, i.e. of paid labour, labelled variable capital (v) by Marx, and surplus labour (value) (s) which was the remainder. Given this framework the proportion of surplus value to value paid for (constant capital plus variable capital) is defined as the (value) rate of profit. This quantity can be expressed as a product of the rate of surplus value (s/v) and the organic composition of capital (c/c + v). Thus, the (value) rate of profit ρ in the ith economic activity
$$ {p}_i=\frac{s_i}{\upsilon_i}\left[1-\frac{c_i}{\left({c}_i+{\upsilon}_i\right)}\right]={r}_i\left(1-{g}_i\right) $$
where ri is the rate of surplus value and gi is the organic composition of capital. But if this were the basis of actual profits, activities with higher proportion of living labour would earn a higher rate of profit (given identical rates of exploitation) relative to one with the lesser labour intensive activity. But since we have to provide for equal rates of profit in all activities, a further step has to be taken to reconcile the theory of unequal value rates of profit with equal actual (or price) rates of profit.

Marx envisaged a pooling of surplus value from all activities at the level of economy and then its redistribution in a transformed form as profits equiproportional to the amount of capital (fixed and variable) invested in each activity. This was done by the price of a product departing from its unit value. The ratio would be above one for activities with organic composition of capital above average and below one for those below average. This condition will reconcile the unequal value rates of profit, given equal rates of surplus value with equal (price) rates of profit. Indeed for Marx this gives a usable rule to predict transfer of surplus value from one sector to another as he did use in his chapter on Absolute Rent (3/XLV).

The problem is however that the numerical example used in 3/IX contained a conceptual error (though this is disputed as we shall see below) which gave the calculations a tentative, halffinished, unsolved appearance. This can be best explained by setting out Marx’s numerical example but in a more general notation. He took five activities labeled i = 1,…,5, each using as inputs constant capital ci and variable capital vi with the gi being different in each activity from the other. The output of the activities were not specifically identified nor was it clear whether they were of the constant capital or the variable capital category. To keep the inputs and outputs separate therefore let input prices be labelled pc, pv and output prices pi.

The value of output can be expressed as
$$ {\displaystyle \begin{array}{l}{y}_i={c}_i+{\upsilon}_i+{s}_i=\left\{\left[1+r\left(1-{g}_i\right)\right]/\left(1-{g}_i\right)\right\}{\upsilon}_i\\ {}\kern0.72em =\left[\left(1+{\rho}_i\right)/\left(1-{g}_i\right)\right]{\upsilon}_i\end{array}} $$
In Eq. (2), we have used Eq. (1) and assumed as Marx did that the rate of exploitation is identical in all activities. (All the variables total value yi as well as ci, vi could be interpreted as being per unit of physical output if thought convenient.) Corresponding to Eq. (2), the price (total revenue) of output was written by Marx as
$$ {p}_i=\left(1+\pi \right)\left({c}_i+{v}_i\right)=\left(\left(1+\pi \right)/\left(1-{g}_i\right)\right){\upsilon}_i $$
Again but especially in this case, variables could be thought of in terms of per unit of output.
To determine π, the actual (price) rate of profit, Marx imposed the condition that the sum of surplus values in all activities was equal to the total of profits over all activities i.e.
$$ \sum \limits_i{s}_i=r\sum \limits_i{\upsilon}_i=\pi \sum \limits_i\left({c}_i+{\upsilon}_i\right). $$
Since however his five units were taken to be of the same size in terms of total value, he also trivially obtained an alternative normalization condition that the total value produced equalled total revenue, i.e.
$$ \sum {y}_i=\sum {p}_i $$
Using the normalization conditions notice that Eqs. (2) and (3) together yield
$$ {p}_i/{y}_i=\left(1+\pi \right)/\left(1+{\rho}_i\right)=\left(1+r\left(1-g\right)\right)/\left(1+r\left(1-{g}_i\right)\right). $$
Thus strict proportionality of prices and values can only hold if either the rate of exploitation is zero i.e. no exploitation or for the case of identical organic compositions of capital gi = g. Given Eq. (4b) it was not difficult to see that the price value differences cancel out in the aggregate. While Marx found some positive and some negative deviations of pi from yi, he had no precise explanation to offer at this stage. It is obvious however as he saw that Eq. (5) implies
$$ {p}_i/{y}_i\gtrless 1\kern1.5em \mathrm{as}\kern0.36em {g}_i\gtrless \overline{g}\kern1.5em \mathrm{where}\kern1.5em \overline{g}=\sum {\overline{c}}_i/\sum \left({c}_i+{\upsilon}_i\right). $$
The problem with Marx’s calculation is not that prices diverge from values; that they must, but that the specification of Eq. (3) is mistaken if Eq. (5) holds. The correct way to write the price equation is to weight the inputs by their respective prices, i.e.
$$ {p}_i=\left(1+\pi \right)\left({p}_c{c}_i+{p}_{\upsilon }{\upsilon}_i\right). $$
At one level, we can see that Marx made a mistake in considering the cost of inputs in value terms rather than in price terms. It has been argued however (Shaikh 1977; Morishima and Catephores 1975) that Eqs. (2), (3), (4a), (4b), and (5) can be thought of as the first stage of an ergodic process. By substituting the values obtained by Eq. (5) into Eq. (3) to modify the input prices, the calculations will converge so that the prices in Eqs. (5) and (3) would be consistent with each other.

But this can only be done if the physical specification of ci and vi is matched to one or more of the commodities produced. If this is not done then we have two more prices than we can solve for. It was Bortkiewicz’s merit to have reformulated Marx’s problem using Marx’s Reproduction Schemes outlined in Capital Vol. 2 to allow for matching specification of physical outputs and inputs with constant and variable capital. This allowed him to reduce the size of the problem (the number of unknowns) and allow for aggregate availability constraints on inputs and outputs. He took a model with three commodities (industries or departments) with Department 1 ‘capital’ good (constant capital), Department 2 ‘wage’ good (variable capital) and Department 3 capitalists’ consumption (luxury) good. Thus, two of his three commodities were inputs as well as outputs in the production process i.e. they are basic in the sense of Sraffa but the third one is an output to be consumed but not an input.

Let the three departments (commodities) be denoted as j = 1,2,3. The value equations are the same as in Marx but Bortkiewicz’s treatment allows a clearer input–output demarcation. Thus, the value equations can be written
$$ {y}_j={y}_{1j}+\left(1+r\right){y}_{2j} $$
where yij is the input of good i in the output of good j etc. The price equations are
$$ {p}_j=\left(1+\pi \right)\sum \limits_i{p}_i{y}_{ij}. $$
Bortkiewicz preserved Eq. (4a) as the normalization condition. But in addition he took care to ensure that the conditions of simple reproduction were satisfied. Thus, he imposed for the two inputs
$$ {y}_i=\sum_i{y}_{ij},\kern1.5em i=1,2. $$
But having implicitly chosen his magnitudes to satisfy Eq. (4b) as well, he imposed a condition
$$ \sum_j{s}_j={y}_3. $$
While Eq. (8) are conditions on total availability of inputs to sustain the required level of output, Eq. (9) is a ‘consumption function’ for the recipients of surplus value. As there is no accumulation by assumption, we require that all surplus value is spent on the ‘luxury good’ produced by Department 3.

Thus Bortkiewicz correctly formulated the problem and even put it in the appropriate general equilibrium framework lacking in Marx’s formulation in 3/IX. The solution is straightforward and need not be given here (see Sweezy 1942, 1949; Desai 1979). This should have settled any debate about the problem. It emerges that prices are systematic functions of values but are not proportional to them. But the solution was published in German in 1907 and did not become generally known until Sweezy described it in his Theory of Capitalist Development, nor did it become available until Sweezy’s translation of it in 1949. Within this forty-year interval, economists’ knowledge of the linear model had advanced as a result of the works of Leontieff and von Neumann. It was obvious therefore that the problem could be reformulated in these terms. Winternitz proposed such a formulation in 1949 and full general solution in terms of n goods was given by Morishima and Seton (1961). Roemer (1980) has shown that the linearity assumption can be dropped and a solution in the ‘Arrow–Debreu language’ can be obtained.

Two areas of controversy arose during the 1970s. First was whether it was necessary to go through the transformation problem at all to solve for prices from physical input–output data. This was raised by Samuelson (1971). Second is a more serious question about the conditions required for solution when there is joint production in the von Neumann–Sraffa sense.

Samuelson’s point can be simply made. In order to arrive at value equations such as Eqs. (2) or (6), we have to translate the data which are in terms of physical output flows and labour inputs into the direct and indirect labour content of inputs. After such a translation, we proceed with the transformation. But as we know from input–output analysis, from the physical input data, one can directly solve for prices from the dual of the Leontieff matrix. If one thought of the purpose of the exercise to provide merely a set of prices consistent with a set of values, he is entirely right. What the criticism misses, however, is that if we were to follow Marx’s purpose in providing a theory of profits, the separation of labour input into paid and unpaid components (which assumes a political economic background) and the use of the concept of the rate of exploitation are required. If one is to reject Marx’s theory of profits, it can be done quite independently of the Transformation Problem, as Wicksteed was able to do even before the publication of Capital Vol. 3 since he rejected the labour theory of value, classical or Marxian, as such (Wicksteed 1884; see Desai 1979, for details).

The second line of criticism is much more serious. This is because it claims that positive surplus value is neither necessary nor sufficient for positive profits i.e. it denies the existence of any mapping from values to prices that can satisfy certain general conditions. The problem is with Marx’s treatment of fixed capital. In his formulation of the value equations, Marx takes a flow measure of non-labour inputs. This suffices if all capital equipment has only one period life since then the stock and flow measures are equivalent. But if the capital equipment lives beyond the production period some account has to be taken of this in writing the value and prices equations. Bortkiewicz was also able to formulate this problem with different rates of turnover of capital i.e. different lengths of life in another, even lesser known, paper of his (Bortkiewicz 1906–7). But he took the rates of turnover to be fixed and known in advance. this is less general than one wishes (see Desai (1979) for a description). Marx can be said to have used implicitly a neoclassical accounting whereby the rental on capital correctly measures its productive contribution. But as Morishima (1973) points out a von Neumann accounting scheme in a ‘joint production’ model is more appropriate.

It was Steedman (1977) who first constructed a numerical example in which there is negative surplus value but positive profit. This is an example of the generic case of nonconvexities which are known to arise in activity analysis (Koopmans 1951). Steedman made it however an argument for abandoning Marxian value theory in favour of a Ricardo–Sraffa formulation. This suggestion has parallels with Samuelson’s suggestion since the detour via labour values can be shown to be misleading in some cases. It has also been pointed out that the non-convexity problem can arise in the Ricardo–Sraffa scheme just as much as in the Marx scheme. Morishima (1973, 1975) has taken the view that all that is necessary is to reformulate the value price problem under joint production with appropriate inequality constraints so that non-negativity of (surplus) values and prices are assured. This would seem the more rigorous formulation. The question does remain however of the behavioural foundations of the mechanism that will ensure that in a capitalist economy, only activities with positive surplus values are chosen.

The transformation problem thus continues to fascinate economists even as they debate its relevance. It formed the basis in Bortkiewicz’s case for an early formulation of a general equilibrium problem in linear terms. It has been argued that it is more appropriate for planning calculations in a socialist economy than in a capitalist economy whose workings it was supposed to illuminate (Samuelson and Weiszacker 1971; Morishima 1973). To Marxists as to their opponents, more important issues such as the moral justification for capitalism seem to be at stake in the solution or non-solution of this seemingly arid technical problem. This is one reason why it will no doubt go on attracting new solutions and new attacks.

See Also


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© Macmillan Publishers Ltd. 2018

Authors and Affiliations

  • Meghnad Desai
    • 1
  1. 1.