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The Marxian Transformation Problem Gérard Duménil+ and Duncan Foley++ Abstract: The origins of the Marxian Transformation Problem lie in the differences between two central abstractions of classical political economy, the labour theory of value and the equalization of the rate of profit through competition among capitals. Marx proposed that these two principles could be reconciled by distinguishing the production of surplus-value through the exploitation of labour in the process of production from the realization of surplus- value through the price system, and claimed that the equalization of the rate of profit could be viewed as a redistribution of a given surplus-value among sectors. Marx's treatment can be seen as a generalization of the discussion of this problem in Smith and Ricardo. Modern criticisms of Marx's discussion claim that the labour theory of value is an unnecessary detour to the determination of prices and profit rates, and that Marx's claims that total value and surplus-value are conserved when prices equalize profit rates is unfounded. The Single- System Labour Theory of Value (SS-LTV) interpretation maintains Marx’s two fundamental claims in the following formulations: (1) the price of the net product is the expression in prices (price form) of the total value-creating labour expanded during the period, and (2) total profits are the price form of surplus-value, determining the value of labour-power as unallocated purchasing power (UPP) on any set of commodities workers can buy from their wages. These properties hold for any set of prices, not specifically prices of production. The various positions in this controversy are illustrated in a mathematical formalization of the circulating capital model of production. Marx's framework: value, surplus-value, prices and competition Marx consistently distinguishes the notions of value and price, in contrast to contemporary economic language, which uses the term 'value' to refer to prices in a situation of general equilibrium, though the use of the term is rather flexible; for example 'value added' is actually the value of net product measured in price terms. For Marx, value is a 'social substance' manifested in economic relations in the 'form' of prices, though prices are not necessarily proportional to values as we will see. Value and surplus-value We first recall Marx's basic concepts (see also Marx's analysis of capitalist production). Central to Marx's framework of analysis in Capital is the labour theory of value (LTV), which defines the value of a commodity as the 'socially necessary' labour time required by its + MODEM-CNRS, Université de Paris X-Nanterre, 200 av. de la République, 92000 Nanterre, France. Email: gerard.dumenil@u-paris10.fr. ++ New School for Social Research, New York. Email, U.S. foleyd@newschool.edu. production, that is, the labour time required by average available techniques of production for workers of average skill. The LTV is central to Marx's theory of exploitation, a term he uses to describe a situation in which one individual or group lives on the product of the labour of others. According to the LTV, when commodities are exchanged through sale and purchase, no value is created. But this principle does not apply to capitalists' purchase of the labour-power of workers. Workers sell their labour-power, that is, their capability to work, to a firm, owned by a capitalist. The buyer uses this labour-power in production to add value to the commodity produced. The value of labour-power is the labour time required by the production of the commodities the worker buys. But the worker can typically work more hours than are on average required to produce this bundle of commodities. For example, the goods the worker can buy may require 8 hours of labour per day, when the labour-day lasts 12 hours. The difference, 4 hours, is unpaid labour time. If an hour of social labour on average produces a value whose price form is $10, 4 hours of unpaid labour time results in a surplus-value whose price form is $40, which is appropriated by the capitalist. The rate of surplus-value is the ratio of unpaid to paid labour time, in this case, 4/8, that is 50%. Two laws of exchange Marx situates his discussion in the context of the distinction made by Adam Smith and David Ricardo between 'market prices' and 'natural prices'. Market prices are the prices at which commodities actually exchange from day to day in the market. Smith and Ricardo, however, regarded market prices as fluctuating (or 'gravitating') around centres of attraction they called 'natural prices'. ('Gravitation' means that the economy is in a permanent situation of disequilibrium, though in a vicinity of equilibrium where natural prices would prevail.) In the above analysis, Marx assumes that commodities tend to exchange at their values (at prices proportional to values), that is, in proportion to the labour time embodied in them. 'Tend' means here that deviations are obviously possible, but that such prices will 'regulate' the market, in the sense that if the prevailing set of prices systematically under-compensates the labour used in the production of a commodity, labour will move to the production of better-paid commodities. As a result, the supply of the under-compensated commodity will decline, and its price will rise. In reality prices would gravitate around values, which would play the role of natural prices in such an economy. This is the commodity law of exchange. In a capitalist economy, however, capitalists buy not only the labour-power of workers (which Marx denotes as variable capital), but also non-labour inputs, such as raw materials, and fixed capital, such as machinery (which Marx denotes as constant capital). If natural prices were proportional to labour inputs, as the commodity law of exchange posits, capitalists using more constant capital per worker than the average would realize smaller profit in comparison to their total capital advanced, that is, lower profit rates. Marx accepts the idea that competition tends to equalize profit rates in various industries, despite differences in capital advanced per worker, which is the capitalist law of exchange. Marx uses the term prices of production to describe a system of prices which guarantee to the capitalists of various industries a uniform profit rate. Capitalists will invest more where profit rates are larger, and conversely in the symmetrical case. They move their capital from one industry to another seeking maximum profit rates, and this movement result in a gravitation of market prices around prices of production. Marx regards prices of production as the centres of gravitation of market prices, and thus the natural prices relevant to a competitive capitalist economy. Is the theory of surplus-value compatible with the theory of competition? The problem is posed of the compatibility of the capitalist law of exchange at prices of production with the theory of exploitation as extraction of surplus-value. Marx's line of argument is that surplus-value is created in production through the exploitation of labour, that is, in proportion to labour expended, but realized proportionally to total capital invested. According to Marx, this separation between the locus of extraction and the locus of realization does not contradict the theory of exploitation so that capitalist competition is compatible with his theory of exploitation through the appropriation of surplus-value from unpaid labour time. To support this argument, Marx presents a pair of tables (1981, chapter 9) showing the redistribution of surplus-value through deviations of price from values proportional to embodied labour times. All variables are measured in hours of labour time, and as a result prices of production are expressed in the same unit. Because Marx's own calculations involve some extraneous complexity (differential turnover rates among sectors), it is more useful to consider the simplified case shown in Table 1. Two industries exist, each of which advances the same capital of 100, but divided in different proportions between the purchase of non- labour inputs (C) and labour inputs (V). All capital is used up during the period, so that the rate of profit is the ratio of surplus-value to total capital advanced, r =s/(c+v). The rate of surplus-value is uniform and equal to 100%. Consequently, surplus-values are equal to variable capitals. Surplus-values and values are computed in each industry. When prices are proportional to values, profit rates differ between the two sectors. Prices of production are determined in Marx's procedure by summing up all surplus-value, a total of 40, and redistributing it in proportion to total capital, that is 20 in each industry, to equalize profit rates on the capitals advanced. Table 1: Industry Constant capitals, C Variable capitals, V Total capitals, K=C+V Surplus- values, S=V Values of commodities produced, Λ=K+S Profits, Π 'Prices of production' of commodities produced, P=K+Π 1 70 30 100 30 130 20 120 2 90 10 100 10 110 20 120 Total economy 160 40 200 40 240 40 240 The procedure illustrates a straightforward 'redistribution' of surplus-value. Clearly, the sum of prices, 240, is equal to the sum of values, and total surplus-value is, by construction, conserved in the form of profit. These observations are expressed in two Marxian equations concerning the entire economy: Sum of values = Sum of prices of production Sum of surplus-value = Sum of profits Note that these compact formulations are not rigorous, since values and surplus-value are measured in labour time and prices and profits in money. Thus, 'Sum of values' should read 'Sum of prices proportional to values'. A simple way out of the problem of units is to use one of these equations to define the general level of prices. For example, the sum of prices of production could be set equal to the number of hours corresponding to the sum of values. Then, Marx's line of argument implies that the surpluses in both sets of prices are equal, as in the second equation. This simple calculation illustrates the idea that profits are 'forms' of surplus-value, that is, unpaid labour. Approximations Marx is, however, aware that the type of computation illustrated in Table 1 is not satisfactory, since the evaluations of constant and variable capital have not been modified despite the fact that prices have changed: 1) When natural prices are prices of production, non-labour inputs are purchased on the market at prices of production, not at prices proportional to values. It is, therefore, not correct to conserve the evaluation of constant capital: 'We had originally assumed that the cost-price of a commodity equalled the value of the commodities consumed in its production. But for the buyer the price of production of a specific commodity is its cost-price, and may thus pass as cost-price into the prices of other commodities. Since the price of production may differ from the value of a commodity, it follows that the cost-price of a commodity containing this price of production of another commodity may also stand above or below that portion of its total value derived from the value of the means of production consumed by it. It is necessary to remember this modified significance of the cost-price, and to bear in mind 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. Our present analysis does not necessitate a closer examination of this point.' (III,9) (References for quotations are to the volume and chapter in Capital. Sources can be found on the internet, in the Marx-Engels Library: http://www.marxists.org/archive/marx/works/, or in Marx, 1976, 1978, 1981.) 2) There is a similar problem concerning variable capital. When commodities exchange at prices of production, workers will not be able to buy the same bundle of commodities with a wage corresponding to a purchasing power expressed, as in Marx's calculation, as a certain number of hours of labour time, as when prices are proportional to values. Marx is also aware of this problem: '[…] the average daily wage is indeed always equal to the value produced in the number of hours the labourer must work to produce the necessities of life. But this number of hours is in its turn obscured by the deviation of the prices of production of the necessities of life from their values. However, this always resolves itself to one commodity receiving too little of the surplus-value while another receives too much, so that the deviations from the values which are embodied in the prices of production compensate one another. Under capitalist production, the general law acts as the prevailing tendency only in a very complicated and approximate manner, as a never ascertainable average of ceaseless fluctuations.' (III,9) It is not easy to understand Marx's position from these notes (which he never revised for publication). It does seem that the analysis requires a 'closer analysis', since the revaluation of constant capital at prices of production will in general make the sum of prices of production deviate from the sum of values, or make the sum of profits deviate from the sum of surplus- values. While it is true that a redistribution of surplus-value through a system of prices of production does not alter the living labour expended in production, so that over the whole economy the deviations from value 'compensate one another', the value of labour-power will remain constant only if workers consume commodities in the same proportion as they are produced in the whole economy, which is implausible. The phrase 'average of ceaseless fluctuations' suggests the averaging out of market prices to prices of production rather than the averaging of surplus-value across sectors. If Marx's use of the term 'approximately' is taken literally, it would appear that the LTV and the theory of exploitation he introduced in Volume I of Capital are only 'approximately' true! Although Marx is conscious of the problem, it is impossible to consider his solution as rigorous. In the formulation of the two equations above, it appears that, when the calculation is done rigorously as in the formal setting below, the second equation does not hold! Later critics have judged this a devastating refutation of Marx's theories of value and exploitation, which in turn has led to ongoing controversy. Earlier approaches The foundations of the transformation problem can be found in the first analyses of competition and prices in capitalism, beginning with Adam Smith and David Ricardo, on which Marx elaborated. The distinction between values and prices remains somewhat fuzzy in these authors. Smith fails to establish a clear relationship between value and profit rate equalization as the principle determining 'natural prices'. Thus, one characteristic feature of these approaches, from which Marx was unable to depart completely, is that two sets of prices (the two laws of exchange above) are considered, one proportional to values (embodied labour times), and the other equalizing profit rates (a dual system), when only one price system prevails in real-world capitalism (a single system): 1) A system of prices proportional to values (embodied labour times) plays a role in the analyses of Smith, Ricardo and Marx. Only Marx, however, clearly distinguishes the two systems from the start. 2) The determination of the 'surplus', when such a concept exists (as in Ricardo and Marx), is posed in the first system and imported into the second, instead of being analysed directly within the second system. This dual system approach lies at the basis of the phrase 'transformation problem', which refers to the transformation from one system into the other. Adam Smith Smith's point of departure is an 'early, rude' state of society, before the establishment of private property in land and means of production. There, Smith contends, products of human labour will exchange in proportion to the labour time required to produce them. Smith offers as an example that if it requires two days on average to kill a beaver, but one day to kill a deer, a beaver will tend to exchange for two deer. Smith's argument supporting this conclusion rests on the assumption that any hunter can choose to allocate time to hunting deer or beaver, so that if the exchange ratio were higher or lower than the labour time ratio, hunters would shift from the under- to the over-remunerated productive activity, and force the exchange ratio back toward the labour time ratio. The viewpoint is clearly that of the commodity law of exchange. Smith applies the same type of reasoning to argue that once means of production have become private property (which he calls 'stock', and later economists called 'capital') the ability of owners to shift their capital from one line of production to another will tend to equalize the profit rate across different sectors of production. The viewpoint is now that of the capitalist law of exchange. David Ricardo Ricardo critiques and corrects Smith's analysis. Ricardo originally based his theories of prices and distribution on Smith's first principle that the labour expended in producing a commodity determines its price in exchange. But Ricardo, elaborating on the dual system approach, examines the necessary quantitative difference between the two principles that might determine natural prices more carefully than Smith. Ricardo understood that the proportion between capitals invested in non-labour inputs and labour is not uniform across industries, and that this fact implies a discrepancy between the two sets of prices, but he regarded these deviations as quantitatively limited. Prefiguring Marx's investigation, Ricardo was concerned to work out the properties of the first system (values) to derive conclusions concerning distribution, which he supposed were also valid in the second system (prices of production): 1) When natural prices are proportional to values (embodied labour times), it is obvious that there is a trade-off between the shares of output which respectively go to workers and capitalists: Workers create all the value added to inputs, and buy a share of output whose production requires less labour time than they expend. In contrast to Smith, Ricardo had a clear view of this mechanism. This division of total output between workers and capitalists was crucial to his analysis, because of its implications in terms of economic policy. (For example, Ricardo was in favour of a low price of corn, which, in his opinion, would increase the profits of capitalists by lowering wages—and encourage capital accumulation). 2) Ricardo would have liked to conserve the straightforward distributional properties he derived from the assumption of prices proportional to values, even while acknowledging the quantitative difference between such natural prices proportional to values and natural prices that would equalize profit rates across industries. But Ricardo understood that, in the profit rate-equalizing system, the natural prices of commodities may change with a change in the real wage (due to the distinct compositions of capital) even if the labour required in production remains unaltered, contrary to what happens in the first system, where values remain unchanged with a change in the wage. Thus, with Ricardo's analysis, we are getting closer to Marx's framework and problems. The rebellious classical