Simultaneous valuation vs. the exploitation theory of profit

Submitted by libcom on July 27, 2005

Simultaneous valuation vs. the exploitation theory of profit
Author: Kliman, Andrew J Source: Capital & Class 97-112 no. 73 (Spring 2001): p. 97-112 ISSN: 0309-8168 Number: 70073393 Copyright: Copyright Conference of Socialist Economists Spring 2001


This paper shows that interpretations of Marx's value theory which value inputs and outputs simultaneously imply that surplus-labor is not the sole source of profit-even in the absence of joint production. Contrary results, such as the Fundamental Marxian Theorem, rely crucially on restrictive and implausible conditions that are shown to be unnecessary for reproduction. In contrast, the temporal single-- system interpretation conforms to the exploitation theory of profit under completely general conditions.


Despite their other differences, all interpretations of Marx's value theory agree that it identifies the exploitation of workers, i.e., the extraction of surplus labour, as the sole source of profit. Proponents of the various interpretations, moreover, all claim to have replicated this feature of his value theory. Yet the mathematics of their systems often tells a different story. As I will show, in those systems in which the prices and values of inputs are determined simultaneously with the prices and values of outputs, the extraction of surplus labour is insufficient and, generally, unnecessary for the existence of positive profit. In these 'simultaneist' interpretations, then, surplus labour is not the sole source of profit.

It is well known that, when joint products are produced, certain specifications of the standard interpretation are incompatible with Marx's theory of profit (see Steedman, 1977). As section II will show, however, all simultaneist interpretations (not only the standard one) are incompatible with his theory, even in the absence of joint production.1

Because theorists have failed to study the problem in a general setting, this incompatibility has not received attention. In some special cases-those in which a positive physical surplus or positive net product of every good is produced in every period-simultaneist interpretations do imply that surplus labour and positive profit go hand in hand. Yet section III will demonstrate that this result cannot be generalized. I will argue, moreover, that these special cases impose conditions that are much more restrictive and less plausible than is usually thought. In particular, economies can easily reproduce themselves physically without satisfying these conditions.

Section IV will show that an alternative, non-simultaneous, interpretation of Marx's value theory does imply that surplus labour is both necessary and sufficient for positive profit, even under the most general conditions. A brief summary and conclusion follow in section V.

Before proceeding, a few methodological comments are in order. At various points, I will note that the attempts to reconcile simultaneous valuation with the exploitation theory of profit rely on unrealistic postulates-positive physical surpluses of all goods, equalized profit rates, etc. These comments are not intended as criticisms of any economic theory or model for a lack of realism. I take no position here on whether a theory's postulates should be realistic. The reason I will discuss realism is instead simply to demonstrate that simultaneist definitions imply that surplus labour is not necessary or sufficient for profit to exist in the real world in which we live. If these demonstrations are valid, they are valid even if it is appropriate for formal theories to employ unrealistic postulates, and even if theorems pertaining to imaginary economies are interesting and useful.

The point is that, whether or not it is appropriate to employ unrealistic postulates for other purposes, it would be logically impermissible to use them to draw deductive inferences about real-world situations. Conclusions that are derived validly from a postulated world may or may not hold in the real world. To determine whether they do hold, one can sometimes test the conclusions empirically. That, however, is impossible in this case. Empirical evidence can tell us whether surplus labour and profit in fact coexist. It cannot tell us whether simultaneous valuation is compatible with the theory that they coexist because surplus labour is necessary and sufficient for profit to exist. This question can only be answered deductively, by ascertaining whether there exist conditions under which simultaneous valuation leads to the contrary conclusion. It is to this task that I now turn.


A. The Fundamental Marxian Theorem

In the standard interpretation of Marx's value theory, distinct price and value systems exist, and the inputs and outputs in each are valued simultaneously. Another distinctive feature of this interpretation is that it construes wages in the price system as the price of the wage goods workers receive, and wages in the value system as the value of these wage goods.

Employing this interpretation, Okishio (1993a, 1993b) discovered a set of theorems that Morishima (1973) later dubbed the `fundamental Marxian theorem' (FMT). The FMT is often said to have shown that surplus labour is necessary and sufficient for positive profit when no joint products are produced (see, e.g., Howard and King, 1992: 230, 239).

Yet some versions of the FMT hold only if all producers' profit rates are equal in every period. This is a very particular case; if profit rates are only approximately equal, or only equalized over a span of time longer than one period (two days instead of one, for instance), these versions of the FMT no longer hold. The analysis below considers instead the general versions of the FMT (e.g., Okishio 1993a: 33; Okishio 1993b: 80-81; Roemer 1981: 47-50), which prove that the theorem holds for any set of positive market prices, not just for 'normal' prices. Yet these versions of the FMT rely crucially on an equally restrictive condition: in every period, a positive physical surplus of every good must be produced.

Physical surplus is output net of both consumed inputs and workers' consumption, and, in this interpretation, profit is simply the vector of physical surpluses valued at end-of-period (replacement) prices. Using the usual input-output notation,2 the column vector of physical surpluses is psi = (I - A - bl)x, so profit is

Unless the net products of all goods are non-negative, the aggregate price of the net product, and thus sigma, can be negative, even in highly productive economies. Imagine that net products of almost all goods are positive and large, and only a few are slightly negative. If the prices of the latter group are sufficiently high, the aggregate price of the net product will be negative. Thus, an economy that would have a positive sigma under certain prices could have a negative sigma under different prices. Even a slight change in prices could lead to such a reversal.

A couple of other perverse implications of these interpretations are noteworthy. When sigma is negative, equation (5) implies that a fall in the money wage rate will lead to a fall, rather than a rise, in the amount of surplus labour extracted. As an anonymous referee has noted, moreover, necessary labour (the labour-time equivalent of money wages) is defined here as (l/sigma) w/x, so it is negative when sigma is negative. Workers supposedly produce an equivalent of their wages in less than no time! No oddity of the labour market or technology underlies this result-workers' wages and the amount of work needed to reproduce their means of subsistence are both positive, and necessary labour might well be positive if only relative prices were different.

All of these paradoxes disclose a serious conceptual flaw in the claim that the monetary expression of the value added by living labour can be measured by the price of the net product.4

The proportionality of surplus labour and profit also fails to imply that surplus labour is necessary for profit to exist. As Dmitriev (1974) discovered, if we imagine a fully automated economy that produces a positive net product of all goods -- and if, in addition, prices in such an economy exist and are positive - then profit as defined above is positive, even though no labour or surplus labour is extracted.

Apart from this case, the interpretations in question do imply that, when the price of the net product happens to be positive, positive profit and positive surplus labour will coexist. The relevant issue, however, is not whether they coexist, but why. Unless a theory denies that profit could be positive if no human labour were employed-and those under consideration seem not to do so-then we must conclude that it admits the possibility of positive profit without surplus labour. Putting the same point differently, the only way to refute Dmitriev's challenge to Marx's theory of profit is to deny that the physical surplus of a fully automated economy is effectively the same thing as profit under capitalism. This requires that one deny either that the price of the physical surplus constitutes profit, or that this surplus could have a positive price under complete 73 (Spring 2001): p. 97-112 The definitions of profit given above do not do so.


Perhaps the main reason that the obvious points made in section II have not received attention is that theorists have been interested in economies that are able to reproduce themselves physically. Negative physical surpluses or net products have been thought to imply an economy incapable of long-run reproduction, and have therefore been ignored.

Yet the appeal to physical reproducibility is either an evasion of the issue at hand or the result of a logical fallacy. Assume for the sake of argument that if an economy is capable of reproduction, then surplus labour and profit as defined in the simultaneist models are both positive (or both negative or zero). It does not follow that surplus labour is either necessary or sufficient for positive profit. Analogously, if I am a man, then I am both male and adult. Yet not all males are adults, nor are all adults males.

In any case, it is simply not true that long-run reproduction requires positive physical surpluses or net products of all goods. All actual economies produce some negative net products, and therefore negative physical surpluses, because some goods (386 computers, for instance) are used as inputs without being reproduced. The economies sustain themselves and even grow by producing, instead, similar but not identical goods (586 computers).5 Yet, as was noted above, simultaneist theorems that surplus labour is sufficient for positive profit do require the postulate that all net products are positive. Since this postulate is violated in every actual economy, it follows that the theorems do not apply to the real world.6

It is impossible, moreover, for simultaneists to construct comparable theorems to cover real-world situations, because simultaneous valuation is impossible when some inputs are not reproduced as outputs. To compute the aggregate price of the net product, one takes the gross price of the outputs and subtracts the replacement cost of the inputs, i.e., the vector of inputs pre-multiplied by their end-of-period prices. Yet inputs that have been used up without being reproduced do not have end-of-period prices, so this is impossible.

One could, of course, use their prices when they entered production, but then one would not be valuing inputs and outputs simultaneously. The only other alternative is to impute end-of-period prices to the inputs by trying to establish an equivalence between them and goods that have replaced them as outputs. Yet any attempt to homogenize heterogeneous things is not only conceptually dubious; it also leads to arbitrary results. One estimate may conclude that the price of the aggregate net product is positive, while another, even slightly different, estimate may conclude that it is negative. The truth of a theorem that surplus labour is necessary and/or sufficient for positive 'profit' would then depend on the idiosyncrasies of the estimators!

Yet even if we ignore non-reproduced inputs, it is very probable that actual economies, even highly productive ones that do reproduce themselves over time, fail to satisfy the received definition of 'reproducibility' (e.g., Roemer, 1981: 19). This definition requires economies to produce non-- negative physical surpluses of all goods in each and every period. As I shall show presently, however, reproducibility actually requires only that non-negative surpluses be produced over some longer time span (and that initial reserve stocks be of sufficient size).

Roemer (1981: 19) first states that reproduction requires that no stock be run down to zero. He notes correctly that one way of 'assuring' this-i.e., one sufficient condition for reproducibility-is to postulate that all physical surpluses are non-- negative in every period. Yet immediately thereafter, he pronounces this postulate a 'requirement' for reproducibility -i.e., a necessary condition. It is easy to show that this is incorrect.

Table 1 depicts a two-good economy in which the production of each good requires 0.4 units of both goods. Due to fluctuations in output levels, a negative net product (and thus physical surplus) of good A is produced during the first hour, and a negative net product of B is produced during the second. Over the course of these two periods, however, 25% more of each good is produced than is used up. Given an initial reserve stock of A of at least 1 unit, there is no technological barrier to this economy's expanded reproduction.

Even though some net products are negative, such an economy satisfies the Hawkins-Simon conditions. In essence, these conditions define a self-sustaining economy as one in which all net products would be positive at some levels of output. And all net products would be positive here if, say, 15 units of each good were produced during each period.

Table 1

Since 15 units of each good are indeed produced over two hours, it is possible to include this case among those in which all physical surpluses are non-negative in every period. One needs only to redefine the length of a period as two hours instead of one. Yet once one does so, simultaneist theorems that surplus labour is necessary and sufficient for positive profit will become false. Market prices may change during the lengthened period in such a way that, for instance, surplus labour is extracted in both sub-periods, but profit is negative in each of them and therefore over the lengthened period as a whole.7 (In the present example, this would occur if the relative price of good A were sufficiently high during the first hour and sufficiently low during the second.)

The simultaneist theorems are therefore true only if a nonnegative surplus of each good is produced in each and every period, no matter how short the period. A period in this context can be no longer than the length of time during which prices remain constant; but they can change from one instant to the next. The shorter the period, however, the less likely it is that all physical surpluses will be positive. Over very short periods, it is almost inconceivable that this is the case. Many factories and offices shut down overnight, but night in one part of the world is midday in another. Some business is therefore always using up some input that its supplier is not reproducing at that moment. Hence, if the theorems in question are formally true, they fail to apply to the real world, because one their crucial premises never holds, while if they do apply to it, they are false.



Due to their static character, simultaneist interpretations of Marx's value theory grant value no role in explaining the dynamics of capitalism. Although some proponents of simultaneist interpretations have acknowledged this, they seem untroubled by it. They contend that the `core of the explanatory power of the labour theory of value lies in the analysis of exploitation' rather than in dynamic analysis (Dumenil and Levy (2000:142)). And, invoking the FMT and similar theorems, they have argued that their interpretations do imply that exploitation of workers is the sole source of profit.

This paper has demonstrated, to the contrary, that simultaneism and the exploitation theory of profit are incompatible. The FMT holds only when all physical surpluses are positive (or profit rates are equal) in every period, and similar theorems pertaining to more recent simultaneist interpretations hold only when all net products are positive in every period-no matter how brief the period. These conditions have been shown to be implausible and completely unnecessary for reproduction. A choice between simultaneous valuation and the exploitation theory of profit must therefore be made.

Marx's value theory thus seems to be far more of a `package deal' than has hitherto been recognized. The attempts to fragment it into dynamic and static aspects, and to reject the former while embracing the latter, have not succeeded. When his value theory is given a static interpretation, not only do Marx's explanations of dynamic issues, such as the tendency of the profit rate, seem to be false, so does his explanation of the origin of profit, a putatively static issue. Conversely, the temporal single-system interpretation, which vindicates the internal consistency of his value theory in other respects, also vindicates the logical coherence of the exploitation theory of profit. One may now in good conscience turn directly to Capital, unencumbered by others' 'corrections' of its alleged errors, in order to help analyze and understand the world in which we live.


1. Because it refrains from asserting any relationship between surplus labour and profit (measured in terms of money or a numeraire), the interpretation of Wolff, Callari, and Roberts (1984) is an exception.

2. A = [a^sub ij^] is a square matrix of input-output coefficients; aij is the amount of good i used to produce one unit of good j. b is a column vector of wage goods per unit of living labour, I is a row vector of living labour requirements per unit of output, and x is a column vector of outputs. I is the identity matrix.

3. When no physical surpluses are negative, but some are zero, and some prices and/or values are zero, the aggregate worth of the physical surplus vector can be zero when valued at prices and positive when valued at values, or vice-versa.

4. See Kliman (1997) for other criticisms of this concept.

5. lam indebted to Alan Freeman for emphasizing this crucial point.

6. An anonymous referee has noted that the case of 386 computers concerns `technological change over time creating obsolescence in durables, and not ...valuation in a timeless world where there are negative net products! The point is presumably that the models used to deduce the FMT and similar theorems disregard this case. I agree. It is precisely for this reason

that the theorems do not apply to the real world. This is true whether or not it is legitimate to abstract from this phenomenon and whether or not the theorems are interesting and useful.

7. This could occur even if prices fluctuate only slightly and the economy is in equilibrium in all other respects-growth rates and profit rates are equalized over the course of the lengthened period, technology is not changing, wage rates are equalized, etc. Numerical examples demonstrating this possibility are available from the author at the Department of Social Sciences, Pace University, Pleasantville, NY 10570 USA or [email protected].

8. Of course, it is not sufficient. The exploitation theory of profit would not hold under a different definition of inflation, or under different temporal conceptions of value added and the monetary expression of labour-time.


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Andrew J. Kliman teaches economics at Pace University in New York. He has written extensively on value and crisis theory