Labor Productivity and Marxist Theory of Labor Value

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Labor Productivity and Marxist Theory of Labor Value

Alejandro Valle Baeza

División de Posgrado, Facultad de Economía, Universidad Nacional Autónoma de México

Av. Universidad 3000 Circuito Interior Fac. Economía Edif.. B México 04510 D.F. phone (5255)56222148, fax (5255)56222158, e-mail: valle@servidor.unam.

I am in debt to Ron Baiman and especially to Takao Fujimoto for valuable commentaries to previous versions of this paper; remaining errors are my own responsibility.

This paper analyzes the relationship between labor productivity and the Marxist theory of value. I shall define aggregate productivity based on the theory of labor value in order to improve our understanding of several aspects of productivity: a) what change of labor productivity is measuring; b) to rigorously derive the relationship between real wages, productivity and exploitation; and c) to lay the foundations for an international comparison of productivity.


This paper analyzes the relationship between labor productivity and the Marxist theory of value. Labor productivity is a notion empirical economists use widely when they develop wage policies or make comparisons between countries, for instance. Theoretical economists: Neoclassical, Keynesians or Radical economists also make use of this notion. In contrast, the Marxist theory of labor value is defended by very few even within the field of Marxism itself. This paper points out that whoever uses the concept of labor productivity, however, is implicitly using the theory of labor value, perhaps in most cases even without knowing. Clarifying this relationship between economic practice and the theory of labor value will certainly benefit both empirical economists and economic theorists.

I shall not attempt to look into the important issue of unproductive labor, a theme of great relevance to our study that has been widely studied above all for its impact on the performance of the rate of profit. Researchs by Gillman (1956) and Moseley (1991) clearly show the relevance of the theme; Laibman (1999) highlights the inherent difficulties.

I shall first define aggregate productivity based on the theory of labor value in order to improve our understanding of several aspects of productivity: a) what change of labor productivity is measuring; b) to rigorously derive the relationship between real wages, productivity and exploitation; and c) to lay the foundations for an international comparison of productivity.

I shall later see how this paper is related to other pieces of research that analyze the social character of productivity, which, for example, appears as a dependent variable of conflict in the work place.1 Lastly, we shall present our conclusions.

I believe our results will be a contribution to the development of heterodox economics by showing that the concept of labor productivity not only has a practical application, but is also theoretically correct within the framework of the theory of labor value even if this is not so within mainstream theory. Thus when mainstream economists use this concept, they might be doing bad orthodox neoclassical economics, but good heterodox economics, even without knowing. Heterodox economic theory may benefit from developments in Marxist theory since it links theoretical aspects with more practical grounds. In fact, both fields must fertilize each other.

I Productivity in Marxist theory

Marx suggested that productivity is the reciprocal of the value of commodities.2 This idea has re-emerged in various Marxist authors, such as Gouverneur (2002: Appendix 4), and Valenzuela (1986: 113). In order to understand this conception, we must not forget that socially necessary labor means first of all that both direct and indirect labors are expended in the production of each commodity. The efficiencies with which the different commodities are produced are therefore interrelated. David Ricardo originally set forth this idea, presented in input-output analysis, when he referred to incorporated labor and by Karl Marx when he spoke of abstract labor. In the production of merchandises, many products are also means of production. This leads to something important, the Marxist definition of productivity implies that an increase in the efficiency with which certain commodities are manufactured has an impact on the production of many other commodities, and might even encompass the whole of production system. The Marxist concept of abstract labor emphasizes that: every single commodity is rigorously the result of applying a portion of social labor and not only of the labor process that immediately appears as its origin. We can see this clearly if we treat value mathematically: Given technical conditions A, value is defined as3

M = L(I-A)-1 (1)


A is the matrix of technical coefficients, aij is the physical amount of commodity i required to produce a physical unit of commodity j.

L row vector of simple labor coefficients

M row vector of labor values

This equation is derived from the definition of value as the sum of past labor or the value of the means of production MA and of direct labor L:

M = MA + L

It can pose what gross output would be needed to produce a unit of final demand X for a commodity, if we represent the final demand vector of one unit of commodity i by Yi and the corresponding gross output by Xi , it can be seen that:

mi =L (I-A)-1Yi = LXi (2)


X is a column vector of gross outputs

Y is a final demand column vector

Yi is a column vector with elements yj = 0 j i, and yi=1

Xi = (I-A)-1Yi

Equation (2) clearly shows the idea that a commodity’s value is the direct and indirect labor expended in production distributed as demanded by technical conditions in production. Equation (2) can also be interpreted as output of commodity i per worker is 1/mi and the potential consumption per worker will therefore also be 1/mi.

Taking the above into account, a commodity’s productivity can be defined as:

лi = 1/mi (3)

assuming, for example, the production of a commodity with a value of a tenth of a working-year; each occupied worker will thus be able to consume up to ten units of this commodity. Productivity limits maximum output per worker (and simultaneously limits maximum consumption per worker) for each commodity. This production or consumption would only be reached if commodity i were the exclusive component of final demand.

In general, we can say that the value of any producible basket Y per worker must be equal to one4:

MY /LX = 1 (4)

We thus suggest that defining productivity is analogous to the case of a commodity

П=1/MY (5)

Units of measure

We shall here only look into the problem of the units with which productivity is measured. We thus assume that vector L is constituted by elements li, simple labor requirements, in working-year per physical output unit V/ui. These must be the units of vector M. It thus follows that productivity лi will have units ui/V. This evidences that the productivity of a industry cannot be compared to the productivity of a different industry, and that only growth rates can be compared. It is meaningless to say that the pin industry is more productive than the auto industry. Aggregate productivity П will have units u1 u2..un /V. Hence, at aggregate level, productivity can only be compared when the same commodity basket is used.

II Productivity variation analysis

We here present a result that shows how to make an approximate estimate of the changes in the aggregate productivity of a country’s whole economy. The GDP per worker is what is called "labor productivity". For the neoclassicals, it is important to note that the measure of labor productivity is partial and rigorously speaking incorrect since it does not consider all "production factors". Baumol , Blackman, and Wolff (1989:227) note that, "It is obvious that total productivity –better known as multifactor productivity - is the best input efficiency index, it would thus seem that labor productivity is a measurement that sheds light on the results of the productive process for its participants."

Why is it obvious that the total productivity of the factors measures input use efficiency? Apparently because when labor productivity is used, changes in the quantity of inputs or in the productivity with which inputs are produced are ignored, since it only takes into account the labor directly spent in production. Equation (6) shows changes in the GDP per employed worker at constant prices:

If we hypothesize that prices are proportional to the values

P = α M

It follows that

where subindex 1 denotes the year of the study, and the variables without subindex correspond to the base year.5

Equation (7) says that a Paasche productivity index is equal to the labor productivity index or to the GDP index per worker at constant prices when prices are proportional to the labor values. This result shows that the so-called labor productivity reflects changes in efficiency of both the labor spent in inputs and directly spent labor. In the former index, each branch’s productivity is weighted by its contribution to the final demand. If a branch does not produce consumption goods or goods that are used for net investment, it will not have any direct influence on average productivity. Changes in the productivity of a means of production affect average productivity since they affect the values of all the commodities that use it as a means of production. Because of this, an increase in productivity in one industry may increase productivity in many other industries or even in the whole economy. This needs further explanation. In the first place, it is a result that shows one of the aspects of the social character of labor: since all industries are interdependent, a change in the productivity of one industry can affect all the other industries. By saving labor in the production of a means of production, more labor can be devoted to production in all the industries that use such merchandises. The effect an increase in productivity may have thus depends on both demand and supply. An increase in productivity in any industry makes it possible:

a) To increase the consumption of the commodity in question and of all the commodities that used such merchandise as means of production, directly or indirectly.

b) To increase the consumption of all the commodities involved among all the workers, depending on the correlation on forces among the classes and to the possibilities of capital accumulation.
The result demands prices are value prices, which imply that errors will occur when changes in productivity are measured by current prices. We here only mention the effects demand has on productivity, the relationships between demand and productivity are broader and more complex as Marx rightly noted in chapter X in book III of Das Kapital.

This definition reflects the social nature of productivity, since each product results from the labor performed by many of the workers not only by workers directly involved. Not only each production process uses a multiplicity of inputs, but also because an increase in the efficiency with which a means of consumption is produced enables an increase in consumption of all workers. Under capitalism, however, this possibility is not necessarily concretized. An increase in productivity in the production of means of production allows a greater consumption of the means of production and of all the commodities that make use of it. Taking this argument further, assuming an increase in the productivity of a mean of production used directly or indirectly in the production of all commodities, the consumption of all commodities would also increase.

III Productivity and real wages.

Based on purely national acounting considerations, it is evident that there is a relationship between wage participation in the GDP (σ), labor productivity (yr), the rate of real wages (sr), the implicit GDP deflator (ID) and the consumer prices index (CPI)6.

we cannot here see the relationship that must be maintained between real wages and labor productivity since both the GDP price index quotient and consumer goods quotient intervene. The index ratio is however often disregarded, perhaps because it is considered that indexes vary in a similar way, as happens in the USA, for example.

For mainstream economy, this vagueness is not so serious as the fact that labor productivity is not multifactor productivity. If the neoclassicals or the Keynesians aimed at coherence, they would use marginal productivity of labor or a similar concept based on their vision of multifactor production. As far as we know, this is non-existent, although we shall see that a Marxist theory has no difficulty in finding a relationship between real wages and productivity.

The first point to note is that iff prices are proportional to values, wage share (σ) is equal to the necessary labor rate (ln), i.e., the quotient between the labor time workers devote to their reproduction and their total labor time.

The ratio of real wages to labor productivity therefore expresses not only the change of the ratio necessary labor / living labor, but also how the productivity of consumer goods sector moves in relation to productivity in the economy as a whole. 7

IV Comparison of productivity levels among countries

If a basket is expressed by column vector Y* and the unitary value magnitudes of the commodities corresponding to country A are expressed by row vector MA, the productivity measured by basket Y* will be:

ПA* =1/ MAY* (10)

We shall use indexes A and B in this section to refer to two countries. In the equation above, ПA* is productivity measured with the values of country A, first index, and basket Y*, second index. According to this notation, we shall have two productivities: ПA* and ПB*.

The law of one price (LOP) holds that commodity prices are equalized through the exchange rate. The LOP is highly reasonable, inasmuch as it is not possible for the price levels between two countries to be too distant from each other because that situation would ruin the production of a large amount of commodities in the high price country. We shall here use the weak version of the LOP that holds that it is only the price levels that are equalized. Based on the previous definitions:

where c*BA is purchasing power parity (PPP), defined in terms of basket Y*, and PA and PB are the row vectors with the prices in both countries.

Prices are considered to be value prices:

Substituting 12 into 11 we have:

Multiplying (12) by YA and dividing it by total employment LAXA we get to:

where yA is the GDP per worker in country A, doing the same for country B and substituting such equations into (13) we get to:

This equation is expressing that the quotient of the GDPs per worker at current prices in a common currency is equal to the productivity quotient so long as two imposed conditions are met: that prices be value prices and the exchange rate be the PPP.

It is important to note that with equation 15 we are not estimating productivity quotient with two final demands of different countries which will be in general incomparable, but we are estimating the productivity quotient with which the same basket Y* is produced in each country. Productivity appears thanks to the purchasing power parity estimate. Since in reality prices are different from value prices, exchange rate c*BA and GDP per worker will only be useful for an approximation to relative productivity.
V The social character of productivity

An abstract definition of productivity does not tell us enough about its meaning. In the sections above, productivity was aggregated for the whole economy or for the sector that produces wage goods together with other highly aggregated variables. Weisskopf et al (1983) and Buchele and Christiansen (1999) explain productivity taking into account variables that I did not include here. What is the relationship between these two visions? The first answer is that they are based on pieces of research that have different levels of abstraction, a different purpose in mind, and turn out to be complementary. Productivity and the surplus value rate are variables that can be explained by other variables since, as Weisskopf et al noted, aggregate productivity is a variable that condenses the actions performed by human beings with “minds and hearts”, capable of struggling and changing working conditions or of being defeated and subjected by the forces of capital to the hardest life conditions. From Marx, who considered that the length of the working day and labor intensity determined the surplus value rate to contemporary authors the surplus value rate and productivity have depended on other variables. Buchele and Christiansen (1999) link changes in productivity with the workers’ rights. Pieces of research such as these are essential to understand productivity and capitalist exploitation nowadays. Our paper aims to help strengthen this line of research in which changes in productivity are measured through changes in output per worker expressed at constant prices. Weisskopf (1983) argue that they would rather use output per worker in order to measure aggregate productivity than a physical measure since the former is applicable to the economy as a whole and the results obtained from both measures ultimately are not that different. Our results show that the Marxist theory of value was right to use this measure of productivity and in doing so it was including another aspect of the social nature of labor: individual commodities result from the application of a portion of the labor of all the workers. The habitual way of measuring productivity through labor productivity considers that each output results from the collective action of a multitude of workers and not only those directly involved. In order to modify productivity, capitalists as a class must see to the hearts and minds of the working class instead of confining themselves to controlling what happens in their own industry.

Another important aspect derived from our piece of research is the relationship between aggregate and disaggregate productivity. To see this relationship, we must remember that output can be measured by final demand or aggregate value, both expressed in constant prices. The reason why this can be done with the equations we have used are obvious since8:

P(I-A)X=PY (16)

The left side of the equation above corresponds to the total aggregate monetary value and the right side to the monetary value of the final demand. Although both methods come up with the same result at an aggregate level, they might nevertheless differ when we try to apply them to a branch or sector. To understand this, let’s analyze a commodity produced with cheaper means of production. In this case, productivity in that industry would increase, but an estimate via aggregate value would not show any changes since there are no changes in the conditions under which such commodity is produced. The method based on

the aggregate value in general underestimates the changes in productivity. Studies for the manufacturing sector, for example, would thus underestimate changes in productivity due to a drop of value in the mining industry or the estimates of variation in productivity of the wage goods sector would be underestimations because they ignore changes in the value of the means of production. Gouverneur (2002, appendix 4) notes this problem and offers an elegant solution to measure changes in productivity at a disaggregate level.

VI Conclusions

Labor productivity is the most widely used way of measuring productivity at an aggregate level and it is used by almost all the economists that conduct empirical studies. We have demonstrated that changes in labor productivity approximately measure changes in reciprocal value of a basket, or that changes in labor productivity, at an aggregate level, measure productivity defined within the theory of labor value: productivity is the reciprocal of value of a basket. Thus defined, productivity changes not only because of savings in both live labor and means of production, but it is also modified by changes in the productivity of these means of production. We also demonstrated that the ratio of real wages to output productivity is an approximate reflection of the necessary labor – live labor ratio multiplied by productivity in the ratio of wage goods to total productivity. Thirdly, we held that the ratio of GDP per worker at PPP in two countries is an approximate expression of the productivity ratio with which these countries produce the basket that is used to estimate the PPP. In the three cases, the measurements turn out to be approximations since in reality prices are not directly proportional to labor values, but more or less differ from them. Finally, we very briefly discussed how even if we explain that productivity variation, as defined in Marx’s theory of labor value as being measured by the changes in output per worker; we would still need to explain what changes in productivity are due to. The research work carried out by the Social Structure of Accumulation (SSA) is indispensable for this explanation. We held that the line of research that explains the social character of productivity and my own line of research are complementary and that we hope this is of use. We suggested that a possible application of our work would be to identify possible errors in measuring productivity when aggregate value at constant prices is used. Other possible applications for more concrete analyses along the lines of the SSA are: understanding the role unproductive labor plays and how foreign trade and price value discrepancies influence the estimation of productivity.


Baumol, W., Blackman, S.A.B. and Wolff, E.N. (1989) Productivity and American Leadership, The Cambridge Massachusetts and London: MIT Press.

Buchele, R. and Christiansen, J. (1999). "Labor Relations and Productivity Growth in Advanced Capitalist Economies," Review of Radical Political Economics 31(1), Winter, pp. 87-110.

Ferguson, C.E. (1971). The neoclassical theory of production and distribution. London: Cambridge University Press.

Foley, D.(1982)"The Value of Money The value of Labor Power and the Marxian Transformation Problem", Review of Radical Political Economics, v. 14(2): 37-47.

Gouverneur, J. (2002) Comprender la economía. Louvain-la-Neuve: Diffusion Universitaire CIACO.

Guillman, J.M. (1956). The Falling Rate of Profit. London: Dennis Dobson.

Laibman, D. (1999) “Productive and Unproductive Labor: A Comment,” Review of Radical Political Economics, Vol. 31, No. 2, June, pp. 61-73

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Valle B., A.(1991)"Productividad: las visiones neoclásica y marxista", Investigación económica, 198, oct.dec.: 45-70.

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1. One of the seminal works underlying this idea is Weisskopf, et al., (1983)

2 . "Real saving economy –saving- consist of the saving of labour time (minimum (and minimization) of production costs); but this saving identical with development of the productive force.” Marx, K. (1973: 711)

3  This value formulation appeared some time ago in the literature. See by example Seton, F. (1957). Michio Morishima (1973) surely contributed most to the dissemination of this presentation. 

4 This is deduced by post-multiplying the definition of value M=L(I-A)-1 by Y, the final demand vector and substituting the definition X=(I-A)-1 into the resulting equation.

5  In order to demonstrate this, substitute equation 4 and the input-output equation X=(I-A)-1Y into equation 6 in order to get to equation 7.

6 This expression is derived directly from definitions of wage share, real wage and labor productivity.

7 This expression is derived by substituting CPI and ID definitions and then in equation 8.

8 This results from the expression: intermediate demand AX plus final demand Y equals total sales X.

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