A Defense of the Fundamental Marxian Theorem

 Introduction: 

The Fundamental Marxian Theorem1 - the proposition that capitalist exploitation is a necessary and sufficient condition for the existence of profits - has been negatively appraised on several grounds.2 Perhaps the most damning indictment is that the proof is tautological; “‘...profit is the source of surplus value’ is as formally valid as vice versa; ‘only if the profit rate R* can be positive… can the rate of surplus value r* be positive…’” (Samuelson 1974, 64). It will be argued that this claim is specious. The first section re-establishes the mathematical proof in Leontief/Sraffa-style notation. The analysis will proceed as follows: technical conditions of production → value magnitudes → complete prices → prices of production → generalized proof. The second section unpacks the rationale behind this procedure. Samuelson’s confusion comes primarily from not understanding Marx’s historical-logical method. There are valid, extra-mathematical reasons to assert that while the Theorem itself is true, Samuelson’s Inverted Theorem is meaningless as a selective description of economic reality. The author agrees with critics that the labor theory of value is unhelpful in describing competitive capitalist price-formation, and that the Transformation Problem is a fool’s errand. But it does not follow that labor values are unhelpful in describing exploitation, or that Marx’s theory can be accurately recast in “neo-Ricardian”3 terms. The last section concludes. 

Notation:

• y - Column Vector of Net Output (Surplus Product)

• q - Column Vector of Gross Output

• A - nxn Technical Coefficients Matrix {aij}

• I - Identity Matrix

• (I - A)-1 - Leontief Inverse (Total Requirements) Matrix
• aL - Row Vector of Labor Coefficients

• c - Column Vector of Labor Consumption Goods

• A(w) - Augmented Matrix

• w - Income for Labor (only a wage when r > 0)

• ej  - jth column of I

• vj  - Labor Value of the jth Commodity

• v - Row Vector of Labor Values

• e - Rate of Exploitation

• p* - Complete Price Row Vector (r = 0)4

• w* - Complete Labor Income

• p - Row Vector of Production Prices (r > 0)

• r - Profit Rate

• λ - Eigenvalue

• xL - Left Hand Eigenvector

• k - Arbitrary Scalar

System of Equations:

Technical conditions of production

1. y = q - Aq = (I - A)q

1. all elements of y > 0, and at least one element (yj) > 0

2. q = (I - A)-1y

3. Total employment = aL (I - A)-1y

Assume advanced Labor Income

then 

4. A(w) = A + caLw

Value-Magnitudes

5. q = (I - A)-1ej

6. v = aL (I - A)-1

6.1. vj = aL (I - A)-1ej

7. e = [v(I - A - caL w)q]/(vcaLqw)

Prices for the Complete Labor Income Rate

8. p* [A + caLw*] = p*

8.1. or [because of the Perron-Frobenius condition λ = 1 ∈ A(w*), p*A(w*) = p*]

Lemma 1: Some scalar k > 0 exists, so p* = kv

Prices of Production

9. pA(w) (1 + r) = p.

9.1. 1/(1 + r) = λ ∈ A(w) ⋀ p = xL ∈ A(w*) → Det[1/(1 + r) I - A(w)] = 0

Take pc = 1 as numéraire, suppose that w < w* and that interindustry organic compositions of Capital are unequal.

The Fundamental Marxian Theorem

Lemma 2: No scalar k > 0 exists, so p* ≠ kv.

Definition: Under both conditions of Lemmas 1 and 2, labor is exploited if and only if e > 0.

Fundamental Marxian Theorem: r > 0 ↔ e > 0.

Explanation: 

We begin by formalizing the technical conditions of production. The matrix of technical coefficients is a non-negative square matrix whose elements show the inter-industry input requirements i for unit outputs in each sector j. The net output vector is the gross output vector net of replacement costs, alternatively termed the physical surplus. The Leontief inverse shows in its columns the total input requirements generated by one unit of output; it functions similarly to the Keynesian multiplier, but is a matrix instead of just a scalar (Tsoulfidis and Tsaliki 2019, 70). The total employment of labor is represented by a vector of labor coefficients, also specifying the physical surplus they are required to produce. Finally, the augmented technical coefficients matrix denotes a real labor consumption vector, which is set to unity and measures relative exchange-values in terms of labor commanded. These steps are all crucial to a rigorous proof of exploitation because the technical conditions of production analytically precede both value-magnitudes and prices of production (Steedman 1977).5

Once these technical conditions - the state of technology, size and composition of output, and consumption of labor - are known, then labor values can be defined as the amount of socially-necessary labor time required to increase net output by one unit of the jth commodity. The rate of exploitation is then defined as the ratio of labor embodied in the physical surplus not included in labor consumption to the labor embodied included in labor consumption. This is only a rate of surplus labor at this stage of abstraction because capitalist relations have not been explicitly denoted in the equations yet, at which point it becomes a rate of surplus value. The extraction of surplus labor in the form of periodic crop yield occurred under feudal modes of production, and under slavery in the form of human property as well. But capitalist exploitation - surplus value extraction - is unique. This is why Petri’s claim, following Garegnani, that Marx’s theory of exploitation is simply an “...explanation of why wages are below their potential maximum,” (Petri 2015, 8; italics not added) as well as his “Genghis Khan” analogy (2015, 5-6) do not completely do Marx justice. Such a conflation of capitalist and pre-capitalist exploitation misses key dynamics which make the real abstraction of social labor time so important in understanding capitalism’s central regulation. The site of exploitation is no longer the production process itself, but rather a distributive injustice stemming from the bargaining advantage of the capitalists. The latter problem definitely exists in Marx, but only the former is unique; the struggle over bigger shares in wages or profits exists in the Classicals also, so its implications can only be so radical.

Exploitation in pre-capitalist forms did not incentivize technological innovation and continuous productivity increases. In the case of feudalism, neither absolute nor relative surplus labor was a continuous subject of maximization, because the landed aristocracy had no direct control over the length and intensity of the working day in their manner of exploitation. In comparing the two, Marx asserts: “The wage form thus extinguishes every trace of the division of the working day into necessary labour and surplus labour, into paid and unpaid labour. All labour appears as paid labour. In the corvée, the labour of the worker for himself, and his compulsory labour for his lord, differ in space and time in the clearest possible way” (Marx 1867, 381; italics added). He later revisits this theme, stressing again that “...this identity of surplus value with unpaid labour of others need not be analysed here because it still exists in its visible, palpable form, since the labour of the direct producer for himself is still separated in space and time from his labour for the landlord and the latter appears directly in the brutal form of enforced labour for a third person” (Marx 1894, 576; italics added). In the case of slavery, absolute surplus labor was certainly maximized but not relative surplus labor, due to the lack of competitive incentives which inspire cost-cutting. The physiological limit to how quickly and how much slaves could work was not appreciated by their owners. But on the contrary, even the dullest capitalist understands that too lethargic and culturally dissatisfied a workforce will, at least in the long run, render their enterprise uncompetitive either due to falls in productivity or rises in worker organization. Thus, although wage labor is certainly immiserated and regularly pushed to its limit, the specific relations of production cause the norm of capitalist exploitation to be revolutionization of the means of production, and thereby the cheapening of the reproduction of labor-power (variable capital). Capitalist exploitation uniquely conceals itself in the apparent justice of the wage contract, wherein labor-power generates more than its reproduction.

As has been well-known since Smith (2000), the price system corresponding to labor receiving the total surplus product is characterized by equality of labor embodied and labor commanded. Relative prices here are proportional to relative labor times. This is sometimes referred to as the condition of the maximum or complete wage rate (Pasinetti 1979), but this is a fetishistic use of terms. Wages are dependent on their opposite, namely profits. To be paid a wage already implies separation from the means of production and the imperative to sell labor-power, which is impossible for a hypothetical wherein direct producers own all inputs and also receive all net income. 

This tedious exposition is all relevant in order to challenge the idea that the proof of exploitation is a mere tautology. Surplus labor extraction is a general feature of class societies, capitalist or not. Surplus products appear in all societies agrarian and forward, which includes potential socialist ones in the future. But the form and function exploitation takes under capitalism - command over the working day, the seemingly voluntary nature of wage contracts, and the imperative to stay competitive - is what gives rise to profits, the necessary form of appearance of surplus value. Surplus labor is nothing new in the history of class society, and in those cases profit only came in a relative form; buying cheap and selling dear. Whereas, its renewed form of surplus value coincides with the advent of profits on production; they did not exist prior to this new form of exploitation. So while the Theorem may seem to run either way, its qualitative determinations prohibit the notion of positive profits being a necessary condition for positive surplus labor. It is not “metaphysical” (Robinson 1957) to include this qualitative commentary. It is a crucial selective process which makes the empirical multiplicity of economic facts reveal their necessary, causal, and human determinations; the social relations of production.6 Samuelson would like algebra to explain more than it is capable of. In the spirit of Coase’s torturing data quip, if you overwork equations long enough, they will say stupid things. He caustically wonders, “Why don't people argue about the ‘meanings’ of Wicksell the way they do about those of Ricardo and Marx?” (1974, 64). He must have missed the dispute between Keynes and Hayek; perhaps when changing the course of 20th century economic thought himself, others doing the same fell out of his periphery. 

The final mathematical step is Sraffian prices of production (Sraffa 1960), the algebraic Rome where all roads lead. Marx’s theory does not require the stringent assumption of perfectly uniform capital intensities in order to be robust, unlike Samuelson’s (Garegnani 1970).7 Even in the expected cases where individual commodity prices do not converge with their labor embodied, exploitation of surplus value plays a vital explanatory role. Without exploitation in its concretely capitalist form, profits on production do not exist. Steedman (1977) is right to say that value-magnitudes are redundant in calculating relative prices; as noted above, both are derived from technical conditions. But he is wrong to say they are redundant altogether, assuming one wants to analyze the social relations and class dynamics which are unique to capitalism and intrinsic to its reproduction. Recasting exploitation in terms of the physical surplus (Garegnani 1984, 2018) conflates it with pre-capitalist forms, which is not only analytically sloppy, but certainly not what Marx meant. The genus of exploitation is the institutional advantage of one group to extract a disproportionate share of society’s surplus labor from the laboring majority, according to Marx. But the species of capitalist exploitation separates the laboring majority from the means of production, and imposes as a necessity the purchase of their capacity to work in order to squeeze out, at a frequent and immediate rate (not just “after the harvest”), as much surplus value as possible. Marx established this as the origin of profits, and this discovery marked a significant difference between him and the Classical Political Economists: Neither Smith nor Ricardo identified the origin of profits, although Smith did acknowledge the superior bargaining power of the capitalists (2000, 75 - 76). Marx's indictment of Ricardo is particularly illuminating: “It is moreover clear that if a certain development of the productivity of labour must be assumed in order for surplus labour to exist, the mere possibility of this surplus labour, that is, the existence of that necessary minimum productivity of labour, does not in itself make it actual. In addition, the labourer must first be compelled to work in excess of that time; and this compulsion is exerted by capital. This is missing in Ricardo, and hence also the whole struggle over the regulation of the normal working day” (Marx 1951, 308; italics added).

Conclusion:

Profit on production requires the exploitation of labor, of a concrete historical form. This is not a reversible identity, as it establishes analytical priority to one variable (e) and logical dependence to the other (r). Nor does it require the simple labor theory of value to be true, as well as any despairing over solving the Transformation Problem. The former Marx knew, but the latter can be fairly described as a logical misstep, with Marx as the progenitor of a hundred-plus year wild goose chase.8 Marx’s failure to transform input prices does not raze his theory of exploitation, which is also not just a story of conflict and bargaining power. Assuming that it is, and that value magnitudes do not contribute to understanding it, blurs modes of production together. The Marxian theory of capitalist exploitation does not stand or fall with the labor theory of prices, although the exploitation process itself must be described in terms of labor-time lest all-important details be neglected. Apologists and critics of Marx alike assume his difficulties in the theory of prices to destroy the entire edifice; the former despairs while the latter rejoices. Both are ill-founded in their judgment. 

Footnotes:

1. The name was given by Morishima (1973), with whom most associate the Theorem. But it was originally arrived at by Okishio (1963). A prototype of the proof can be found in von Charasoff (1909, 1910), who also anticipated the Shaikh/Bródy/Morishima iterative solution to the Transformation Problem. 

2. Other arguments also challenge the Theorem on different terrain, in ways which the author is not currently concerned with. One is the problem of joint production, where positive profits can coexist with zero or even negative surplus value (Steedman 1977, Petri 1980). For rejoinders to these charges, refer to Sheppard and Barnes (1980, 51 - 54), as well as Lucas and Serrano (2017). The other popular claim is that labor is arbitrarily chosen as an exploited input, so that in systems with basic goods one can demonstrate a “commodity exploitation theorem” (Bowles and Gintis 1981, and Roemer 1981, 1982). For answers to this curiosum, see Basu (2020) as well as Yoshihara and Veneziani (2013). 

3. I use this term here and not “Sraffian” because in other ways, Sraffa can arguably be placed in the Marxian tradition itself. The significance of his Standard Commodity (Sraffa 1960, 18-33) is that it demonstrates, contrary to Bortkiewicz, that the distribution of income can be physically ascertained before knowing relative prices. Sraffa is therefore a “succesivist” like Marx, and not a “simultaneist” as is so often assumed (Kliman 2006, 76 - 77). “Neo-Ricardian,” although originating as a slur from Marxists against followers of Sraffa (and his predecessors like Dimitrev and Bortkiewicz), captures the essence of argumentation coming from many in that camp, sympathetic to Marx, who think physical Sraffa prices alone can salvage Marx’s theory of exploitation.

4. Shaikh (1977, 1984, 2016) calls these “direct prices.” They are prices proportional to labor embodied, in two main special cases. The first is when r = 0, and labor income costs are all that must be covered. The other is in the case of equal organic compositions of capital, because surplus value is proportionally distributed to each industry as to not require a change in prices to maintain the uniformity of r. In all other cases however, “At no other wage-level do values follow a simple rule” (Sraffa 1960, 12).

5.  Shaikh (1981, 1982) argues against this by saying that the technical conditions of production are themselves determined by “...the labor process. It is human productive activity, the actual performance of labour, that transforms ‘inputs’ into ‘outputs’, and it is only when this labour is successful that we have any ‘physical production data’ at all” (Shaikh et al. 1981, 280). The author wholeheartedly agrees, but this is only relevant insofar as the profit rate is a positive function of the length and intensity of the working day. When it comes to relative prices, calculating direct and indirect labor must still start from the physical coefficients, and in this sense Steedman is correct. 

6. “The point of such descriptions can be illustrated with a somewhat different but related distinction captured in descriptive statements of the type: 'Michelangelo made this statue of David.' Note that the description is remarkably selective on facts: it says nothing about the tools and instruments used in making the statue; it is silent on the ownership of the huge block of marble that Michelangelo used; it eschews the patronage that Michelangelo received. The description is not based on the assumption that Michelangelo would have been able to make the statue even without these other things, but on the contrast between the role of Michelangelo in the making of the statue and the role of these other things. In going from all possible factual statements about a phenomenon to a pithy description, there is, in a sense, a loss of information, but there is also, in another sense, a gain of focus” (Sen 1978, 177; italics added).

7. Karma showed no mercy to Samuelson, as he discovered that the exact stringent condition which would salvage the labor theory of prices was the same condition he needed in order for his “pseudo-production function” to work. He displayed a lost art of humility in immediately acknowledging his errors, and thanking his interlocutors for calling him on them (Samuelson 1966, Samuelson and Levhari 1966). 

8. Still, the empirical strength of the labor theory of prices is a puzzle in this respect. The author finds claims of statistical artefacts or spurious correlation unconvincing. But data itself, while important, cannot establish causal laws; it can only give us an object of investigation.

Works Cited:

Basu, D. (2020). Exploitation of Labour or Exploitation of Commodities? Economics Department Working Paper Series, 297. https://doi.org/10.7275/20622939

Charasoff (von) G. (1909). Karl Marx über die menschlishe und kapitalistiche Wirtshcaft. University of California Libraries.

ー (1910). Das System des Marxismus. Berlin: H. Bondy.

Garegnani, P. (1970). Heterogeneous Capital, the Production Function and the Theory of Distribution. The Review of Economic Studies, 37(3), 407–436. https://doi.org/10.2307/2296729

ー (1984). Value and Distribution in the Classical Economists and Marx. Oxford Economic Papers, 36(2), 291–325. http://www.jstor.org/stable/2662883

ー (2018). On the labour theory of value in Marx and in the Marxist tradition. Review of Political Economy, 30(4), 618-642. https://doi.org/10.1080/09538259.2018.1509546

Gintis, H., & Bowles, S. (1981). Structure and practice in the labor theory of value. Review of Radical Political Economics, 12(4), 1-26. https://doi.org/10.1177/04866134810120040

Kliman, A. (2006). Reclaiming Marx's capital a refutation of the myth of inconsistency. Lexington Books. 

Levhari, D., & Samuelson, P. A. (1966). The nonswitching theorem is false. The Quarterly Journal of Economics, 80(4), 518-519. https://doi.org/10.2307/1882912

Lucas, G. D., & Serrano, F. (2017). Understanding and overcoming the “positive profits with negative surplus-value” paradox. Brazilian Journal of Political Economy, 37, 587-604. http://dx.doi.org/10.1590/0101-31572017v37n03a07

Marx, K. (1867). Capital: A Critique of Political Economy, Volume I: The Process of Production of Capital. Progress Publishers, Moscow, USSR. https://www.marxists.org/archive/marx/works/download/pdf/Capital-Volume-I.pdf

ー (1894). Capital: A Critique of Political Economy, Volume III: The Process of Capitalist Production as a Whole. International Publishers, NY. https://www.marxists.org/archive/marx/works/download/pdf/Capital-Volume-III.pdf

ー (1951). Theories of Surplus Value. Lawrence and Wishart, London, UK.

Morishima, M. (1977). Marx's Economics: a Dual Theory of Value and Growth. Cambridge University Press.

Okishio, N. (1963). A mathematical note on Marxian theorems. Weltwirtschaftliches Archiv 91 (2).

Pasinetti, L. L. (1979). Lectures on the theory of production. Macmillan. 

Petri, F. (1980). Notes and comments - Positive Profits Without Exploitation: A Note on the Generalized Fundamental Marxian Theorem. Econometrica (pre-1986), 48(2), 531. https://www.jstor.org/stable/1911114

ー (2015). On some modern reformulations of the labour theory of value. Contributions to Political Economy, 34(1), 77-104. https://doi.org/10.1093/cpe/bzv003

Robinson, J. (1957). An essay on Marxian economics. Macmillan. 

Roemer, J. E. (1981). Analytical Foundations of Marxian Economic Theory. Cambridge University Press.

ー (1982). A General Theory of Exploitation and Class. Harvard University Press, Cambridge, MA.

Samuelson, P. A. (1966). A Summing Up. The Quarterly Journal of Economics, 80(4), 568–583. https://doi.org/10.2307/1882916

ー (1974). Insight and Detour in the Theory of Exploitation: A Reply to Baumol. Journal of Economic Literature, 12(1), 62–70. http://www.jstor.org/stable/2721867

Sen, A. (1978). On the labour theory of value: some methodological issues. Cambridge Journal of Economics, 2(2), 175–190. http://www.jstor.org/stable/23596406

Shaikh, A. (1977). Marx's Theory of Value and the Transformation Problem. The Subtle Anatomy of Capitalism. Jesse Schwartz (ed.), Goodyear Publishing Co. 

ー (1981). The Poverty of Algebra. The Value Controversy. New Left Books and Verso, London.

ー (1982). Neo-Ricardian economics: a wealth of algebra, a poverty of theory. Review of Radical Political Economics, 14(2), 67-83.

ー (1984). The transformation from Marx to Sraffa. Ricardo, Marx, Sraffa, 43-84.

ー (2016). Capitalism: Competition, Conflict, Crises. Oxford University Press. 

Sheppard, E. S., & Barnes, T. J. (1990). The Capitalist Space Economy: Geographical Analysis after Ricardo, Marx and Sraffa. Routledge. 

Smith, & Cannan, E. (2000). The Wealth of Nations / Adam Smith ; introduction by Robert Reich ; edited, with notes, marginal summary, and enlarged index by Edwin Cannan. Modern Library.

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

Steedman, I. (1977). Marx After Sraffa. Verso Books.

Tsoulfidis, L & Tsaliki (2019). Classical Political Economics and Modern Capitalism: Theories of Value, Competition, Trade and Long Cycles. Springer Link.

Yoshihara, N., & Veneziani, R. (2013). Exploitation of Labor and Exploitation of Commodities: A “New Interpretation”. Review of Radical Political Economics, 45(4), 517-524. https://doi.org/10.1177/0486613412475192