Transformation problem

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The transformation problem is a problem in classical and Marxian economics. It appears in the work of Ricardo where he notes a difficulty in equating values, as given by his labour theory of value (LTV), to prices – the problem there being that given reasonable assumptions about competition, differences in the proportion of labour to non-labour costs between industries will skew the value-price relation between industries. This kink in the LTV lay unresolved by Ricardo and his immediate successors

In somewhat more detail, the problem is that, according to the LTV, the value of an industry’s output is proportional to the labour that the industry uses; but equalisation of the profit rate by competition demands that the price of its output be proportional to the labour and non-labour capital that it uses (since the rate of profit, π = [profit]/[all costs].) So an industry with a high proportion of non-labour costs (a `capital-intensive’ industry) for example, must either sell its output above value or have a subnormal profit rate. Non-labour capital costs apparently drive a wedge between value and price. Hence the LTV and profit-rate equalisation are apparently in conflict.

Attention to the transformation problem greatly intensified after the publication of volume III of Marx’s Capital in 1894 because chapter nine of that book contained what was taken by some to be a solution to the Ricardian problem. Marx held that in economy-wide aggregate, price equals value, but that in particular sectors or industries output might sell at above or below value. Specifically, in `capital intensive’ industries, it would sell at above value and in labour-intensive industries it would sell at below value. Importantly, he saw this as amounting to a transfer of surplus value from the labour-intensive to the `capital intensive’ sector; the labour-intensive sector does nor `realise’ all of the surplus value that it produces, while the `capital-intensive’ sector realises not only all of the surplus value it produces, but the unrealised portion from the labour intensive sector as well.

Anti-Marxists, noteably Böhm-Bawerk, quickly pounced on perceived flaws in Marx’s analysis, characterising it as a failed defence of the unworkable LTV, and contending furthermore that the entire structure of Marxian economic theory could not stand because it incorporated the LTV as a premise. The Marxian defence to this lay in two directions. One accepted the general conceptual framework of the Böhm-Bawerk-type criticisms, but provided solutions to the transformation problem that were mathematical improvements over Marx’s. This has resulted in solutions that are, arguably, adequate, although so far none meet all of the restrictive conditions that might ideally be applied.[1] The other line of defence was to challenge Böhm-Bawerk’s conceptual framework, claiming that he had not understood what Marx was trying to do in Chapter IX of Capital III. This line holds that Marx’s purpose there was not to provide a theory of price determination at all, but to investigate the post-production distribution of surplus value throughout the economy – an investigation made important by the fact that this distribution tends to obscure, or mystify, the original creation of surplus value by labour during the production process. This investigation is seen to occupy much of Capital III; in Chapter IX, proponents of this line contend, price shifts are considered by Marx only as one means by which surplus-value transfers might be accomplished. Viewed in this light, the value-price transformation is of merely auxilliary significance to the over-all economic theory, and any sketchiness in Marx’s mathematics of the value-price transformation cannot be considered to strike at fundamentals. This line of defence is associated with a Hegelian-dialectical reading of Marx which sees Capital volume I as presenting concepts such as value and surplus value at a very abstract level that employs particular examples only as representatives of society-wide aggregate behaviour, whereas Capital volume III is at a more concrete level of analysis where some particulars are introduced – the important ones for this discussion being different industrial sectors and differing proportions between labour and non-labour capital – and the effects of these particulars in modifying, or imposing further ‘determinations’ on, the abstract principles is considered. Regarding the LTV, it is argued that although in Capital volume I Marx sometimes, à la Ricardo, equated prices with labour times, this was only done as a simplifying assumption at the most abstract level, and, as the later volumes make clear, he did not think that real prices are determined only by labour times, not even on long-term average: in short, Marx did not hold a Ricardian-type LTV.[2]

A more recent criticism of Marx, noteably by Joan Robinson and Sraffians such as Ian Steedman (1977), is that although it is possible to adequately compute prices using the Marxian transformation-problem method, that method is needlessly roundabout because it is possible to go directly from specifications of physical quantities to prices by using, eg., the mathematical techniques of Sraffa, without going through the intermediate step of considering values at all. On this view, value is an ‘unneccessary detour’, a redundant concept that would be best forgotten about. This of course would leave a large body of Marxian discourse looking rather pointless. Against this it is argued that, although the concept may not be needed for the determination of price, it remains useful for understanding other dynamics of capitalism, for linking quantitative economics to considerations of how humans expend their life energies, and for linking economics to ethics.[3]

A still different hue is cast on the debate by Goregnani (1991), who believes there is an epidemic among readers of Marx, including most of the partisans on both sides of the debate just mentioned, which consists in their thinking that the concept of value was given a prominent place by Marx because of its sociological and philosophical implcations. Goregnani says that on the contrary, Marx employed the concept of value because of its importance to a technical issue in classical economics. At that time the technique of simultaneous linear equations, to say nothing of Sraffa’s methods, had not yet arrived in economics, and thus the result established by them that the wage and the rate of profit are not independent but that, other things being equal, a rise in one will result in a fall in the other, was not well recognised. Adam Smith’s method of adding up factor costs to obtain price, for example, treated them as being independent. Ricardo, and then more firmly Marx, introduced value as a means of demonstrating their interdependent, antagonistic, relation.

  • 1 Ricardian problem
  • 2 Bortkiewicz-Sweezy solution
  • 3 Makoto Itoh solution
  • 4 Notes
  • 5 Further reading
    • 5.1 On paper
    • 5.2 Free on the internet

Ricardian problem

Competitive assumptions:

  1. Assume that competition for investment causes the rate of profit to be the same in all industries.
  2. Assume that labour mobility causes the hourly wage to be the same in all industries.

Price of production of commodity:

p = c + W + s     Where: p = price of production of commodity
                         c = non-labour costs of capitalist
                         w = wage costs of capitalist
                         s = gross profit

Rate of profit:

       s
π = ------- = same in all industries
     c + w

Combinig above formulae:

p = (c + w)(1 + π)
p = (c + w)(k1)     where: k1 is a constant = 1 + π

 
Result A:

p = k1·c + k1·w
---------------

But, by labour theory of value,
value-added is proportional to time worked,
and by assumption of equal hourly wage,
time worked is proportional to wage, so:

n α w          where: n  =  value added
n = k2·w              w  =  wage
                      k2 =  a constant

Value of commodity:

v = c + n


Result B:

v = c + k2·w
------------

Comparing results A and B we see that price (p) and value (v) are given by different formulae that are incommensurable except in very particular cases (such as k1 = k2 = 1 or c = 0). In the realistic case of positive profit, k1 and k2 will both be greater than one, which will cause prices to be relatively high, compared to values, in industries with relatively high non-labour costs, and prices will be relatively low in labour-intensive industries.

This result is at variance with the simple assumption, used elsewhere by Ricardo, that values are related to prices by a simple proportionality constant that is the same in all industries. The discrepancy was noted by Ricardo but he did not offer a solution.

Bortkiewicz-Sweezy solution

Started by L. von Bortkiewicz (1868-1931) and revived by Paul Sweezy in his The Theory of Capitalist Development (1942). Their contribution is to convert the cost prices (ci + vi) in each department i from values, which is the form Marx left them in, to prices. They do this by introducing three unknowns, x, y, z which designate the ratio of price of production to value of output, in each of the three departments I, II, III respectively. This leads to a system of three equations:

I   c1x + v1y + r(c1x + v1y) = (c1 + c2 + c3)x

II  c2x + v2y + r(c2x + v2y) = (v1 + v2 + v3)y

III c3x + v3y + r(c3x + v3y) = (s1 + s2 + s3)z

Note: c is constant capital; v, variable capital; s, surplus value; r, rate of profit.

There are three equations but four unknowns (x, y, z, r). To obtain a solution the theorists set z = 1. For convenience the following notations are introduced:

σ = 1 + r

f = c/v

g = (c + v + s)

Expressed using these notations the solution is:

σ = ( -(f2g1 + g2) + [(f2g1 + g2)2 + 4(f1 – f2)g1g2]1/2 ) / 2(f1 – f2)

Sweezy gave the following numerical example:

Value calculation
Department of production Constant capital (ci) Variable capital (Vi) Surplus-value (si) Value of products (ai)
I.  Producers goods 225 90 60 375
II. Necessary consumer goods 100 120 80 300
III. Luxury consumer goods 50 90 60 200
Total 375 300 200 875

Given the above numbers, it can be calculated that, σ = 5/4,  r = 25 percent,   x = 32/25,   y = 16/15. From this is calculated the prices (as distinguished from the values) of each item in each department:

Price calculation
Department of production Constant capital (cix) Variable capital (viy) Profit (pi) Price of product (Pi)
I.   Producers goods 288 96 96 480
II. Necessary consumer goods 128 128 64 320
III. Luxury consumer goods 64 96 40 200
Total 480 320 200 1000

Source: information is as reprised in Makoto Itoh, The Basic Theory of Capital (1988), pp 211-213.

Makoto Itoh solution

Makoto Itoh has adressed some problems still remaining after the Bortkiewicz-Sweezy solution (and the debates following it) by making careful and explicit distinction between the form of value, which is exchangeability, or `the request of commodities to be exchanged’, and the substance of value, which is embodied labour time. This results in careful separation of the value and price domains.

Itoh gives a numerical example consisting of three tables of numbers (rather than Bortkiewicz-Sweezy’s two). His first table is the same as in the Bortkiewicz-Sweezy (B-S) example, using the same numbers, except that he is very explicit – leaves no ambiguity – that the units in the table are labour time (millions of hours), not, say, a monetary or quasi-monetary unit; and he labels the table `substance of value produced (ai)’. His second table is like the B-S second table, except Itoh chooses a value of z = 1/2 rather than 1, so that all numbers in this table are exactly half those in the B-S second table. He is explicit that the units in this table are money (millions of dollars); and he labels the table `prices of production (Pi)’.

The big difference between Itoh, on one hand, and Bortkiewicz and many other commentators on the other, is this: They see the failure of a B-S type of solution to satisfy Marx’s agregation conditions, namely, that the sum of the values equals the sum of the prices, and the sum of the surplus-values equals the sum of the profits, to indicate more or less serious flaws in Marx’s theory. Itoh, however, regards the non-satisfaction of those conditions as entirely unremarkable, unproblematic, and in fact to be expected, since value and surplus-value are in one domain, and price and profit are in another domain which is only determined in a rather elastic way by the first. What Marx was getting at with his agregation conditions, but failed to spell out clearly enough because he himself was somewhat lax domain-wise, not always maintaining clear separation, is, according to Itoh revealed in a third table which can be constructed, that shows the substance of value acquired in each of the three departments.

Substance of value acquired (a’i)
Department of production ci vi s’i a’i
I 225 90 96 411
II 100 120 64 284
III 50 90 40 180
Totals 375 300 200 875

Note that the table is in the value domain; the units are millions of hours. The ci and vi numbers are the same in this table as in table 1, which they must be since they are technologically determined. The surplus values (s’i), however, are different, since they are the surplus values acquired, or realized, by the capitalists in each department, not, as in table 1, the surplus-values produced. The values-of-products acquired (realized), being the sum a’i = ci + vi + s’i, are also different than in table 1. But now look at the bottom row, the totals. They are the same as in table 1 all the way across: c, v, s’ and a’. Here, says Itoh, is the agregate invariance that Marx sensed. It is not that the sum of the values is the same as the sum of the prices, but that the sum of the values produced is the same as the sum of the values acquired; and it isn’t that total surplus-value is the same as total profit, but that the total surplus-value in production is the same as the total surplus-value acquired.

Notes

  1.  Early solutions were Dmitriev (1898) and Bortkiewicz (1907). Seton (1957) is a mathematically rigorous extension of Bortkiewicz. Overviews of this line are common in general works on Marxian economics such as Howard and King (1976, introduction) and Foley (1986, chap. 6).
  2.  Baumol (1991) makes the case that Marx was interested in distribution of the surplus, not price determination. Fine (2012) says that Ricardo’s use of the LTV was ‘instrumentalist’ (ie., for determining prices) but Marx’s was non-instrumaentalist.
  3.  Albritton (2007), especially chapter 7, ‘Ethics and Political Economy’.

Further reading

On paper

Asterisk (*) denotes material used directly as a source in writing this article.

* Albritton, Robert (2007), Economics Transformed. London, England.

* Caravale, G A, ed. (1991), Marx and Modern Economic Analysis, vol 1. Edward Elgar, Aldershot, Hants, England.

* Baumol, William J (1991), ‘Wages, virtue and value: what Marx really said.’ In Caravale (1991). The article derives from Baumol’s ‘The Transformation of Values: What Marx Really Meant’, Journal of Economic Literature, 12, March 1974; and ‘On the Folklore of Marxism’, Proceedings of the American Philosophical Society, 123, April 1979.

Bortkiewicz, L von (1907), ‘On the correction of Marx’s fundamental theoretical construction in the third volume of “Capital”‘, in Paul Sweezy, ed, Karl Marx and the Close of his System; New York City, USA, 1966.

* Bradley, Ian and Howard, Michael, eds. (1982), Classical and Marxian Political Economy. London, England.

Dmitriev (1898), ‘The Theory of value of David Ricardo’, in D M Nuti, ed., Essays on Value, Competition and Utility. 1974. Pp 37-95.

* Fine, Ben (2012), ‘Labour theory of value’, in Fine, Ben and Saad-Filho, Alfredo, The Elgar Companion to Marxist Economics. Cheltenham, England, 2012.

* Foley, Duncan (1986), Understanding Capital. Cambridge, Massachusetts, USA.

* Goregnani, P (1991), [Sorry, Communpedia contributor forgets the article’s title and didn’t write it down], in Caravale (1991).

* M C Howard and J E King, eds. (1976), The Economics of Marx. Harmondsworth, Middlesex, England.

* Marx, Karl (1894), Capital, volume III. Free copy at marxists.org

Ricardo, David (1817), The Principles of Political Economy and Taxation. Free copy at marxists.org

*Saad-Filho, Alfredo (2012), ‘Transformation problem’. In Fine, Ben and Saad-Filho, Alfredo, The Elgar Companion to Marxist Economics. Cheltenham, England, 2012.

Samuelson, Paul (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. This paper started a noteable debate about the transformation problem in the journal over the next few years.

* Seton, F (1975), ‘The “Transformation Problem”‘, Review of Economic Studies, vol 24. Reproduced in Howard and King (1976).

Steedman, Ian (1977), Marx after Sraffa. London, England.

Sraffa, Pierro (1960), Production of Commodities by Means of Commodities: Prelude to a Critique of Economic Theory. Cambridge University Press.

Free on the internet

  • Gerard Dumenil and Dominique Levy papers. jourdan.ens.fr
  • Gerard Dumenil and Duncan Foley, ‘The Marxian Transformation Problem’.
  • International Working Group on Value Theory valuetheory.org  Some of their session papers are published free on the web, others not.
  • Ernest Mandel and Alan Freeman, Ricardo, Marx, Sraffa e-book.
  • Eduardo Maldonado-Filho ‘Release and Tying up of Productive Capital and the “Transformation Problem’.
  • Fred Mosely Papers http://home.mtholyoke.edu/~fmoseley/

Paul Samuelson

  • * 1970 ‘The “Transormation” from Marxian “Values” to Competitive “Prices”: A Process of Rejection and Replacement’ Proceedings of the National Academy of Sciences of the U.S.A. September; 67(1): 423-45.
  • 1971 ‘A New Labor Theory of Value for Rational Planning Through Use of the Bourgeois Profit Rate’ Proceedings of the National Academy of Sciences of the U.S.A. June; 68(6): 1192-1194.

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