Robert L. Vienneau

Updated June 2004


This article presents a Sraffian interpretation of Marx and investigates how much of the Labor Theory of Value is logically viable. The interpretation is presented by means of a simple two good example, thus restricting to arithmetic the mathematics needed to follow this exposition. The model upon which this example is based, however, easily generalizes to N goods, and derives from work done decades ago by John Eatwell.


Consider a simple capitalist economy that produces only two goods, corn and ale. Assume the amounts of corn and ale produced each year are given, as well as the production processes used in each industry. Corn and ale are each produced by processes that require a year to complete. These processes require a certain number of workers to be hired at the beginning of the year, as well as the purchase of certain quantities of corn and ale to be used as inputs in production. Operating these processes then produces certain quantities of outputs of corn and ale for use at the end of the year. Table 1 shows the amount of inputs per unit output for both industries. The data allow for surplus production, that is for more corn and ale to be produced than are used as inputs.

Labor 1 Person-Year1 Person-Year
Corn 1/8 Bushel3/8 Bushel
Ale 1/16 Bottle1/16 Bottle
OUTPUTS 1 Bushel1 Bottle

Although these data are given in physical terms, production conditions should not be thought of as reflecting purely technical relationships. They also embody social relations, including elements of class struggle. For example, Table 1 might implictly rely on assumptions about the length of the working day, the intensity with which laborers work, how often breaks are allowed, and other elements of general working conditions.

Another point of contention is the role of labor time in the data. Different concrete activities are required in producing corn and ale. Since labor is measured in a single unit, person-years, these differences have been abstracted from, as is indeed the case when labor power is sold on the market.


3.1 The Calculation of Labor Values

Marx claimed that labor values reveal certain fundamental characteristics of Capitalism, especially as regards the exploitation of labor. Before this claim can be investigated by means of the above example, the labor values embodied in corn and ale must first be determined. Three equivalent methods of calculating labor values from the physical data are presented here.

3.1.1 A System of Equations

The first method of calculating labor values postulates that labor is embodied in corn or ale in their production. For example, the labor embodied in corn is the sum of one person year and the labor embodied in 1/8 bushels of corn and 1/16 bottles of ale.The production process for ale yields a similar relationship. These relationships are expressed in Equations 3-1 and 3-2:

        1  +  (1/8) vc  +  (1/16) va  =  vc                (3-1)

1 + (3/8) vc + (1/16) va = va, (3-2)

where vc and va are the labor values of a bushel of corn and a bottle of ale, respectively. This system of two linear equations in two unknowns has a unique solution. A bushel of corn embodies 1 13/51 person years, and a bottle of ale embodies 1 29/51 labor years.

3.1.2 Vertically Integrated Subsystems

The second method of calculating labor values is to imagine rearranging the data to reflect vertical integration of corn and ale production. The economy is assumed to produce a given amount of corn and ale and to require given quantities of corn and ale as inputs in production, leaving certain net quantities of corn and ale available. For the corn industry, group that portion of the corn and ale industries needed to replace the corn and ale used up in producing the net surplus of corn. On a per bushel basis, this vertical integration results in Table 2:

Labor 1 3/17 Person-Years4/51 Person-Year
Corn 5/34 Bushel3/102 Bushel
Ale 5/68 Bottle1/204 Bottle
OUTPUTS 1 3/17 Bushels4/51 Bottle

Notice that the net output of this combination of industries is one bushel of corn. The total labor requirements are 1 13/51 person years per bushel.Thus, this method of calculation yields the same labor value for corn as the first method.

Table 3 shows similar results of vertically integrating the ale industry. Here the net output is one bottle of ale, and the labor requirements are 1 29/51 person years, as expected.

Labor 8/17 Person-Year1 5/51 Person-Years
Corn 1/17 Bushel7/17 Bushel
Ale 1/34 Bottle7/102 Bottle
OUTPUTS 8/17 Bushel1 5/51 Bottles

3.1.3 Reduction to Dated Labor

The third method is only presented schematically and only for calculating the labor value of corn. Begin by imagining the current technique of production has been used forever in the past. The production of one bushel corn requires the inputs of 1/8 bushels corn and 1/16 bottles of ale, as well as one person year. Replace the material inputs of corn production by their own inputs. That is, 1/8 bushels corn and 1/16 bottles ale, purchased at the beginning of the given year, required inputs of 5/128 bushels corn, 3/256 bottles ale, and 3/16 person years, all available a year before the given year. Continue forever this process of replacing produced inputs by the inputs used in their producion. This method will result in an infinite stream of labor inputs, all properly dated. The given year's corn output with the given technique requires labor inputs extending back to Adam and Eve. Table 4 presents the first few terms in this series.

1 Bushel
1/8 Bushel1/16 Bottle 1 Person-Year
5/128 Bushel3/256 Bottle 3/16 Person-Year
19/2048 Bushel13/4096 Bottle 13/256 Person-Year
1 13/51 Person-Years

The (finite) sum of the infinite series of labor inputs illustrated by Table 4 is the labor value embodied in a bushel of corn. This sum turns out to be equal to the labor value for corn already calculated in either of the other two methods. This method can also be applied to ale, resulting in the correct answer as well.

3.2 Exploitation

Now that labor values have been defined for this simple example, Marxian exploitation can be explored. Some further assumptions are necessary. Assume that the workers are paid at the end of the year, that they immediately consume all of their wages, and that they spend them so as to buy three bushels of corn for every bottle of ale. This is a very special proportion for the example. Table 5 shows the inputs and outputs for a single person year expended in producing wage goods.

Labor 3/4 Person-Year1/4 Person-Year
Corn 3/32 Bushel3/32 Bushel
Ale 3/64 Bottle1/64 Bottle
OUTPUTS 3/4 Bushel1/4 Bottle

The gross output of wage goods per person years is 3/4 bushels corn and 1/4 bottles ale. The "constant capital" needed to produce this output is 3/16 bushels corn and 1/16 bottles ale, leaving a net output of 9/16 bushels and 3/16 bottles. Let w denote the proportion of this net output paid to the workers in the form of wages, where w ranges from zero to unity. The remainder stays in the hands of the capitalists in the form of profits.

Since the labor values of corn and ale are known, the physical quantities corresponding to capital, wages, and profits can be evaluated as labor values. The labor value, C, of the constant capital is given by Display 3-3:

        C  = (3/16 Bushels) (64/51 Person Years per Bushel )
                +  (1/16 Bottles) (80/51 Person Years per Bottle)

= 1/3 Person Years (3-3)

The labor value of goods consumed out of wages, called "variable capital" by Marx and denoted by V, is given in Display 3-4:

        V  =  (9/16) w (64/51)  +  (3/16) w (80/51)

= w Person Years (3-4)

The labor value of the goods remaining in the capitalist's hands after replacing the means of producing and paying out wages, "surplus value" S, is given by Display 3-5:

        S  =  (9/16) (1 - w) (64/51)  +  (3/16) (1 - w) (80/51)

= ( 1 - w ) Person Years (3-5)

Notice that the labor value of gross output, 1 1/3 person years, is the sum C+V+S.

The capitalists running firms in the wage good industry only end up with any goods remaining after paying their costs if the wage is less than unity. This means that although laborers work for a full year, the goods they buy out of their wages embody less than a person year. This is Marxian exploitation. An important parameter in Marxian thought is the "rate of exploitation" e. The rate of exploitation is defined by Equation 3-6:

                 e  =  S/V  =  ( 1 - w)/w                               (3-6)

The first volume of Capital was largely devoted to explaining how it can come about that workers are exploited. Why is it that the workers buy goods with their wages embodying less labor than they expend in earning them?

Marx's answer revolved around the distinction between "labor power" and "labor." What the worker sells is not so many hours of labor time, but the ability to work for that amount of time, this latter commodity being known as labor power. Like all commodities, labor power has a value. In this case the labor value is the labor needed to produce the goods the workers consume to maintain themselves so as to be able to work for the desired period. Once they have purchased the commodity labor-power, the capitalists obtain its use value, which is so many hours of labor. The secret of exploitation under capitalism, according to Marx, is the difference between the use value of labor power, that is to say the labor hours expended in production by the workers, and the labor value of labor power, the number of hours needed to produce the goods the workers consume.

Exploitation under Capitalism is perfectly consistent with unconstrained trade, as Marx knew full well:

This sphere...within whose boundaries the sale and purchase of labour-power goes on is in fact a very Eden of the innate rights of man. There alone rule Freedom, Equality, Property, and Bentham. Freedom, because both buyer and seller of a commodity, say of labour-power, are constrained only by their own free will. They contract as free agents...Equality, because each enters into relation with the other as a simple owner of commodities, and they exchange equivalent for equivalent. Property, because each disposes only of what is his own. And Bentham, because each looks only to himself.

Nevertheless, Marx thought the workers are exploited.

The rate of profits in terms of labor value terms is the ratio of surplus value to the expenditures layed out at the beginning of the production period. Since this model, in (sometime) contrast with Marx, assumes workers are paid at the end of the year, the rate of profits in value terms is merely the ratio of surplus value to constant capital:

           pi  =  S/C  =  3 ( 1  -  w )  =  3 e/(1 + e)                 (3-7)

Equation 3-7 is the last relationship in the system of labor-values to be examined here.


No agent in this model is conscious of labor values. Capitalists do not try to maximize the labor value of their profit. Nor do workers try to maximize the labor value of their wages. Capitalist and worker alike worry about price. So the question arises in what sense, if any, can exploitation as described by the system of labor values cast insight on price relationships?

Uniform prices, wages, and rate of profits cannot be expected to prevail at any given time. Some buyers of corn will be paying a higher price than others, and the same will go for ale. Some workers will be getting higher than the going wage and others less. Some firms will have an unusually high rate of profit. These differences in a competitive market will engender a kind of leveling process. Prices of corn, ale, and labor power will tend toward a uniform value in all markets. Similarly, one rate of profit will provide a center of gravitational attraction for the market rate.

We can imagine a price system associated with our physical data where this leveling process has been completed. These "prices of production" or "cost prices" can be thought of as centers of attraction for the observable "market prices." Prices of production are such that they are unchanged at the end of the production period. They also allow the system to reproduce itself. These two conditions, when imposed on the physical data, result in the system of equations given by Equations 4-1 and 4-2:

        [ (1/8) pc  +  (1/16) pa ] (1 + r)  +  w  =  pc                 (4-1)

[ (3/8) pc + (1/16) pa ] (1 + r) + w = pa, (4-2)

where pc is the price of corn, pa is the price of ale, w is the wage, and r is the rate of profit. This system embodies the assumption that workers are paid at the end of the year. The use of the same symbol for the wage as in the labor value analysis implicitly assumes that the numeraire is the net output of the "standard industry," that is 9/16 bushels corn and 3/16 bottles ale. The adoption of this numeraire imposes an additional equation:

                  (9/16) pc  +  (3/16) pa  =  1                         (4-3)

Equations 4-1, 4-2, and 4-3 provide a system of three equations in four unknowns. They can be solved for three of the unknowns, say prices and the rate of profit, in terms of the remaining unknown, the wage.

4.1 The Solution Prices

Prices of production of corn and ale in terms of wages are given by Equations 4-4 and 4-5:

                  pc  =  64 / [ 3 (20  -  3 w) ]                        (4-4)

pa = 16 (8 - 3 w ) / [ 3 (20 - 3 w) ] (4-5)

Suppose wages consume the total net output. So w = 1, and the workers are not exploited. Then prices are 1 13/51 dollars for a bushel of corn and 1 29/51 dollars for a bottle of ale. These are also the labor values of corn and ale. Given different organic compostions of capital among industries, labor values provide centers of attraction for market prices if and only if the workers are not exploited.

Generally, the workers will not be paid the whole output. For wages less than unity prices of production will deviate from labor values. In general, there is no regular pattern to these movements. As the wage falls, prices of production can rise and fall in a very complicated fashion. Does this mean prices are unrelated to the labor embodied in goods? Not exactly. Rather, prices of production are dependent on the whole time stream of labor inputs, not just the total labor value. Section 3.1.3 showed how to reduce the physical data to a stream of dated labor inputs. Prices of production are the sum of the wages paid out to the workers for these labor inputs weighted by the rate of profits appropriate for each particular date. Equations 4-6 and 4-7 express this proposition mathematically:

     pc  =  w  +  (3/16) w (1 + r)  +  (13/256) w (1 + r)   +  ...    (4-6)

2 pa = w + (7/16) w (1 + r) + (25/256) w (1 + r) + ... (4-7)

The problem with a simple labor theory of value as a theory of price is that prices do not merely depend on the total labor embodied in commodities. Rather, the entire time distribution of labor inputs is essential. Since these distributions vary among different industries, the prices of production associated with different levels of wages will be different. Essentially, the problem is one of time. This observation does not make me a proto-Austrian as regards capital theory.

4.2 An Alternate Formulation

Instead of deriving prices of production from the physical data, they can be found by "transforming" the value system. The physical data must be restated in value terms to begin this algorithm. Table 6 restates the physical data in Table 1 such that the output of each industry is one person year of output.

Labor 51/64 Person-Year51/80 Person-Year
Corn 51/512 Bushel153/640 Bushel
Ale 51/1024 Bottle51/1280 Bottle
OUTPUTS 51/64 Bushel51/80 Bottle

Table 7 expresses the data in Table 6 using the labor values of corn and ale found in Section 3. Some Marxists consider this data as primary, not the physical units shown in Table 1.

Labor 51/64 Person-Year51/80 Person-Year
Corn 1/8 Year Corn3/10 Year Corn
Ale 5/64 Year Ale1/16 Year Ale
OUTPUTS 1 Year Corn1 Year Ale

The standard commodity can also be expressed in value terms. Recall that the wage is measured in units of 9/16 Bushels corn and 3/16 Bottles ale. In other words, the wage is w units of 12/17 Years corn and 5/17 Years ale. This completes the labor value data that must be transformed into prices of production.

Let xc denote the "transformation factor" for corn. In other words, xc shows the price of production for a unit of corn whose labor value is one person year. Similarly, let xa denote the transformation factor for ale. The system of prices of production is now expressed by transforming all components of the inputs and outputs of each industry, including labor power, in each industry. As before, assume wages are paid at the end of the year and that a uniform rate of profit prevails in each industry. Equations 4-8 and 4-9 show the resulting system.

        [ (1/8) xc  +  (5/64) xa ] (1 + r)  +  
              (51/64) [ (12/17) xc  +  (5/17) xa ] w  =  xc             (4-8)

[ (3/10) xc + (1/16) xa ] (1 + r) + (51/80) [ (12/17) xc + (5/17) xa ] w = xa (4-9)

These equations can be used to find the ratio of the transformation factors in terms of the wage:

                      (xc/xa)  =  5 / ( 8 - 3 w )                       (4-10)

Notice once again that relative prices are not proportional to labor values. Instead the transformation depends on the value of labor power, a parameter that does not appear in Table 7.

So far we have determined relative values of the transformation factors. Absolute values are found by introducing a "normalization" or "invariant." A number of such normalizations are possible, and Marx incorrectly thought several were equivalent. The one most in the spirit of this essay is to assume that the standard commodity is transformed to a price of unity. Accordingly, assume Equation 4-11 holds:

                    (12/17) xc  +  (5/17) xa  =  1                      (4-11)

The (absolute) transformation factors are then found as in Equations 4-12 and 4-13:

              xc  =  17 / ( 20 - 3 w )                                  (4-12)

xa = 17 ( 8 - 3 w ) / [ 5 ( 20 - 3 w ) ] (4-13)

Given the labor values of corn and ale and their transformation factors, the prices of production of a bushel of corn and a bottle of ale can be calculated. For example, the price of a bushel of corn is found by first calculating how many person years are embodied in a bushel, and then transforming those person years to a price. Equations 4-14 and 4-15 show the results of applying this procedure to corn and ale.

         pc  =  vc xc  =  64 / [ 3 (20  -  3 w) ]                       (4-14)

pa = va xa = 16 (8 - 3 w ) / [ 3 (20 - 3 w) ] (4-15)

Not surprisingly, these results agree with the previous formulation in Section 4.1.

4.3 Prices and Values Compared

The price of production system in either formulation allows one to establish a relationship between the rate of profits and wages:

                    r  =  3 ( 1  -  w )                               (4-16)

Because of the choice of numeraire, the rate of profit is a linear function of wages. In fact, the rate of profit in the price system, given by Equation 4-16, is equal to the rate of profit in terms of labor values (see Equation 3-7 above). Thus, the rate of profit in the price system is an increasing function of the rate of exploitation of labor:

                    r  =  3 e/(1 + e )                                (4-17)

This observation draws one connection between labor values and prices, thereby supporting the assertion that labor values reveal something fundamental about the capitalist system. Some other interesting comparisons are shown by examining the employment of one person year in the production of the standard commodity (Table 8), which, by assumption, is the wage good.

Gross Output
(3/4 Bushel, 1/4 Bottle)
1 1/3 Person-Years$1 1/3
Constant Capital
(3/16 Bushel, 1/16 Bottle)
1/3 Person-Year$ 1/3
Net Output
(9/16 Bushel, 3/16 Bottle)
1 Person-Year$1
Variable Capital
(w 9/16 Bushels, w 3/16 Bottles)
w Person-Years$ w
Surplus Or Profits (1 - w) Person-Years$ (1 - w)

In the production of the standard commodity, total prices equal total labor values, and total profits equal total surplus value. It is as if profits are generated by the exploitation shown in the labor value system. They are redistributed such that each industry gains profit in proportion to their outlay, not according to the amount of labor directly employed. This redistribution results in prices of production that deviate from labor values. The mathematics associated with the standard commodity suggests that the labor theory of value may have some validity when treated as a theory of exploitation.


This article has outlined an argument suggesting that market prices are attracted by prices of production and these prices, in turn, are a veil over essential exploitative features of capitalism.

A critic might assert that this conclusion requires two properties of the example to be true in general:

The critic might then object that these properties only hold true in the example because only the production of the standard commodity was considered in drawing up Table 8. In general, neither gross output, net output, nor constant capital can be expected to be composite commodities in the same proportions as the standard commodity.

The critic might further note that the assumption that wages are in the standard proportion is also perfectly arbitrary. Generally, workers will not necessarily spend their wages in the standard proportions.

The critic might object to the claim that profits result from surplus value. Causation can be read either way. Perhaps workers are exploited because profits are positive; it is not that profits are positive because workers are exploited. Furthermore, corn and ale are exploited too. Labor power can be considered as produced by the goods consumed by the workers, and the physical data can be used to calculate "corn values" or "ale values" as well. What is so special about the exploitation of labor?

How can these criticisms be met? It seems labor is an input unlike other inputs. Neither corn nor ale needs to be motivated to work. Corn and ale cannot consciously resist direction. Nor can they be persuaded to work with a greater intensity. The arithmetic may not fully formalize the relationship of an employee to his employer. But, arguably, the above calculations reflect that relationship.

The critic seems to be wrong in claiming that the above example depends on wages being spent on the standard commodity. The interesting properties of the above model required the wage be measured in units of the standard commodity, that the standard commodity be the numeraire. And this particular numeraire is picked out by properties of the data given in Table 1.

There is another aspect of these models that may justify the Marxist theory of exploitation, the Fundamental Theorem of Marxism. Profit is positive in the price system if and only if labor is exploited. That is, pick any composite good for the wage good. Modify Table 5 so the net output of one worker is in the newly given proportions. Labor is exploited if some of this net output is retained by the capitalists. Similarly, change Equation 4-3 to use this new wage good as the numeraire. It is still the case that the rate of profit is positive in the prices of production system if and only if labor is exploited in the labor values system. But other properties of the model cease to hold. The rate of profit will generally not be a linear function of the wage, and the equalities shown in Table 8 will usually be violated.

Perhaps the interesting properties of the example will usually be approximately true in actual capitalist economies. The standard commodity is in some sense a commodity of average capital intensity, of an average organic composition of capital. No reason has been given to expect consumer goods to be systematically biased in one direction or another from this average. Likewise, investment cannot be expected to be systematically biased. So, although the equalities shown in each row of Table 8 will not generally hold exactly, they will usually be approximately true. Interestingly enough, remarks in Piero Sraffa's unpublished papers suggest that this empirical defense was close to this own opinion.

Consider these criticisms and counter-criticisms. Neither side contends that labor values explain relative prices directly, and Marx never intended to assert relative prices tend to be proportional to labor values. One could accept the critic's objections, discard labor values, but retain a focus on objective data. The Sraffian data are the conditions arising in production and an external specification of the distribution of income. Does this approach provide a methodology consistent with the materialist conception of history and class struggle? If so, it does not suffer from problems in the labor theory of value.

How many have made it this far without long ago consuming a bottle of ale?

To express myself in Terms of Number, Weight or Measure; to use only Arguments of Sense, and to consider only such Causes, as have visible Foundations in Nature; leaving those that depend upon the mutable Minds, Opinions, Appetites and Passions of particular Men, to the Consideration of others.
                                          -- William Petty

Though the earth, and all inferior creatures, be common to all men, yet every man has a property in his own person: this nobody has any right to but himself. The labour of his body, and the work of his hands, we may say, are properly his. Whatsoever then he removes out of the state that nature hath provided, and left it in, he hath mixed his labour with, and joined it to something that it is his own, and thereby makes it his property.
                                          -- John Locke

One point which emerges from out analysis of the classical exchange value and real value theory is that Marx's theory of surplus value is not the result of a 'gross misunderstanding.'...Marx was right in saying that his surplus value theory follows from the classical theory of real value...Moreover, Marx was not the first to draw radical conclusions from it. All pre-Marxist British socialists derived their arguments from Adam Smith and later from Ricardo. Economists did not welcome these inevitable conclusions...He thus touched upon a sore point of economic theory and, probably for this reason, caused so much irritation amonst economists. They often tried not so much to prove him wrong, which would not have been too difficult, as to show that he was an utter fool, a bungler, misguided by those despised German philosophers...The classical theory of value leads inevitably to a rationalist radicalism, if not necessarily in Marx's formulation, at any rate in that direction. For the historian of thought the real puzzle is why the classics did not draw these radical conclusions.
                                          -- Gunnar Myrdal

The 'transformation algorithm' is precisely of the following form: 'contemplate two alternative and discordant systems. Write down one. Now transform by taking an eraser and rubbing it out. Then fill in the other one. Voila! You have completed your transformation algorithm.'
                                          -- Paul Samuelson

It can scarcely be overemphasized that the project of providing a materialist account of capitalist societies is dependent on Marx's value magnitude analysis only in the negative sense that continued adherence to the latter is a major fetter on the development of the former.
                                          -- Ian Steedman

The propositions of M[arx] are based on the assumption that the comp[osition] of any large agg[regate] of commodities (wages, profits, const[ant] cap[ital]) consists of a random selection, so that the ratio between their aggr[egate] (rate of s[urplus] v[alue], rate of p[rofits]) is approx[imately] the same whether measured at 'values' or at the p[rices] of prod[uction] corresp[onding] to any rate of s[urplus] v[alue].

This is obviously true, and one would leave it at that, if it were not for the tiresome objector, who relies on hypothetical deviations: suppose, he says, that the capitalists changed the comp[osition] of their consumption (of the same aggr[egate] price) to commod[itie]s of a higher org[anic] comp[osition], the rate of s[urplus] v[alue] would decrease if calc[ulated] at 'values', while it would remain unchanged at p[rices] of p[roduction], which is correct? - and many similar puzzles can be invented.

(Better: the cap[italist]s switched part of their consumption from comm[oditie]s of lower to higher org[anic] comp[osition], while the workers switched to the same extent theirs from higher to lower, the aggr[egate] price of each remaining unchanged...)

It is clear that M[arx]'s pro[position]s are not intended to deal with such deviations. They are based on the assumption (justified in general) that the aggregates are of some average composition. This is in general justified in fact, and since it is not intended to be applied to detailed minute differences it is all right.

This should be good enough till the tiresome objector arises. If then one must define which is the average to which the comp[osite] should conform for the result to be exact and not only approximate, it is the St[andard] Comm[odity]...

But what does this average 'approximate' to? i.e. what would it have to be composed of (what weights sh[oul]d the average have) to be exactly the St[andard] Com[modity]?

i.e. Marx assumes that wages and profits consist approximately of quantities of [the] st[andard] com[modity].

                     -- Piero Sraffa (quoted in Bellofiore 2001)