the sense that they would definitively disappear, because they are only transformed in the course of this process.

These explanations are not esoteric ones, and we do not want to split hairs on this question, but they are necessary in order for the concepts used to have a connection with the real world. In the real world indeed, final consumption is definitely a destruction. This is obvious for all the non-durable goods, like for instance food, for which there is simultaneity between consumption and destruction, this destruction being also a complete disappearance. But this is also the case for durable goods, even if this destruction can be a very long process for certain goods: in the end there is a complete impossibility for such a durable good to continue to be used as it was initially and during its whole lifespan.

On the contrary, in the case of intermediate goods, like all the raw materials, they most often disappear in the production process in their initial form, but only to reappear under another form at the end of this process, which means literally that they are not destroyed, but only transformed. This is so because their substance is still there, incorporated in the final goods of which they have become an element or a part. Furthermore, in a value-creating process, it is logical for the value of these intermediate goods as a dimension attached to them, not to disappear, but to be transferred to the value of the goods in which they become incorporated.

Evidence of this can be derived from the fact that at the end of the lifespan of a durable consumption good it is more and more frequent that it can be dismantled and recycled, which allows for many of the raw materials which they were made of, i.e. the intermediate goods, to reappear and be transformed again in new production processes, where they are reincorporated into new goods. As for the consumption goods they have been definitively destroyed by this recycling, which confirms their theoretical status of final goods. It must be emphasized that this recycling happens obviously in a completely new production cycle.

Within the production process itself, there are often some tailings, scraps or residues from these intermediate goods, and most of the time, in particular each time that these intermediate goods are valuable, they do not disappear but are recovered to be put back in the process, rather than being wasted. Therefore the word "destruction" is clearly inappropriate to name what happens to intermediate goods in the course of the production process, since they do not disappear completely, but on the contrary can reappear either during this process or at the end of the lifespan of the final consumption goods in which they are incorporated. It is only their original shape or form which are partially modified (rather than destroyed), to be transformed into other ones.

All this explains that final consumption goods cannot at all be treated conceptually like intermediate goods, since these last ones are not consumed but only transformed in the production process, and as such are not the object of an "intermediate" or "productive" consumption, but of a productive transformation. Therefore final consumption goods should not in any case be put on the left-hand side of the equations of the production system, which
represents what enters into the production process, even if these goods are “means of subsistence”, which can constitute as we have seen a part or the totality of “wage-goods”.

Let us conclude that the analysis performed in this very first chapter of *Production of commodities by means of commodities*, devoted to “production for subsistence”, is seriously flawed. Indeed there is no surplus, and therefore there should be no production, according to Sraffa’s own definition of production. Even if one thinks that there is a kind of notional surplus, in the form of subsistence goods which would be both on the left and on the right-hand side of the equations, this would be contradictory to the demonstration, which has just been carried out, that consumption goods by their very nature are final goods which come out of the production process and as such can never enter into this process. This should be kept in mind when we now go further into the analysis of the concept of production in *Production of commodities by means of commodities*.

### 3. Production with a surplus

It is a little ironic in this context to observe that it is in the following chapter of *Production of commodities by means of commodities*, entitled “Production with a surplus” that Sraffa writes that the introduction of a surplus makes “the system becomes self-contradictory” (Sraffa, 1960, p. 6). But this is because the allotment of the surplus, which is supposed to be made by a uniform rate of profit, cannot be determined before the prices, and *vice-versa*. However Sraffa quickly resolves the difficulty by explaining that in fact both are determined simultaneously by the same mechanism.

The interesting thing in this chapter is rather that it shows that the emergence of the surplus has one effect which is the appearance, as part of this surplus, of a new class of products, “which are not used, whether as instruments of production or as articles of subsistence, in the production of others”. As we mentioned previously, Sraffa names them “luxury goods”.

Building upon the demonstration performed in the previous section, it can immediately be observed that, strictly speaking, this definition of luxury goods applies in fact to all consumption goods. This follows because we have shown that the fact that a consumption good be a subsistence good does not in itself transform it into an intermediate good, which belongs to a totally different category of goods, those that can be put on both sides of the equations representing a production process. It follows from this fact that as far as final consumption goods are concerned, no surplus can be determined as a difference between the quantities of these goods which are on the right-hand side and the quantities supposed – wrongly, to be on left-hand side of the equations.

To be sure, there are some particular goods or services which can be in the same physical form, both means of production in the form of intermediate goods (circulating capital), and final consumer goods. They are mostly fluids (water, energy), or some services. But electricity itself does not produce electricity: we know that this is not its use in the production process that makes it reappear for a larger amount at the end of this process. Even in this not
so usual case, the total quantity of this particular kind of goods or services produced during a
period totally disappears during the same period either as intermediate consumption or as final
consumption, without therefore the apparition of a surplus or net product.

In this respect, and to be more precise, even the example of wheat is misleading, because in
the real world wheat in the form of seeds (as an input) is increasingly a processed product
which has undergone some specific treatments, or even has been genetically modified etc.,
and therefore is different from wheat as a pure consumer product. This last one is itself eaten
only after being transformed (ground, conditioned, etc.), and this in a way which makes it a
different product from wheat which has just been harvested. Let us also point out that for most
agricultural products other than cereals, grains or seeds are in any case very different from
harvested products.

Having arrived at this stage, and putting aside fixed capital for the time being, to adhere to
Sraffa's approach, we must understand that there is no surplus either for intermediate goods:
certainly the quantities of intermediate goods used in the production of these intermediate
goods themselves are smaller than the total quantities of these goods which are produced. But
it is obviously because the rest of these intermediate goods are used in the production process
of final consumption goods or fixed capital. Moreover, in a self-replacing state, where Sraffa
repeatedly locates his theory, there cannot be any place for stockpiles of intermediate goods,
which are produced in the exact quantities needed for both their own production and the
production of final goods. Therefore in a given period the total quantity of each of the
intermediate goods which enters the production process, and appears on the left-hand side of
all the equations of the production system using this intermediate good, is exactly equal to the
quantity which comes out of the system, and appears on the right-hand side of the equation of
the industry producing this good in the same system.

**4. The introduction of fixed capital**

To begin with the definition of fixed capital, given at the beginning of Chapter X of
*Production of commodities by means of commodities* (§73), Sraffa does not elaborate a lot,
and limits itself to writing that:

> "...we shall regard durable instruments of production as part of the annual intake of a process, on the same footing as such means of production (e.g. raw materials) as are entirely used up in the course of the year; while what is left of them at the end of the year will be treated as a portion of the annual joint product of the industry." (Sraffa, 1960, § 73, p.75).

There is therefore no doubt that fixed capital is considered in this book as a particular kind, or
a sub-species, so-to-say, of intermediate goods, with the only difference that the life of the
components of fixed capital is longer than the life of intermediate goods, which disappear (are
transformed) in the production process at the same time as they transfer their value to the
goods produced. Fixed capital is indeed made of "durable instruments", or machines, which
implies that their use in the production process is a progressive one which lasts up to the end of their lifespan, during which they progressively transfer their value, up to the end of the production process in which they enter. The only difference with circulating capital seems therefore for Sraffa only a question of time, having to do with the longer durability of their lifespan and therefore of their participation in a production process. If one wonders why time is so important, the only coherent explanation is that the rate of profits is defined not only as a percentage, but as a percentage per unit of time (in fact, the year).

At a conceptual level, it is however extremely confusing to restrict the differences between both types of capital to just this single element of time. Indeed, as we have already pointed out earlier, intermediate goods participate in the production process in such a way as they are effectively and entirely transformed in this process, where they can no longer be found under their initial form at the end of it, because their material substance itself has been incorporated in the final goods which they have contributed to produce. In other words they enter into the process, but do not come out of it.

One must immediately recognize that this is not at all the case for fixed capital goods, which in the real world do not participate in the same way in the production process, whatever their durability and independently of the duration of this participation. Indeed fixed capital goods never disappear in the production process, where their true role is not to be incorporated in the structure of final goods, but on the contrary to participate in the transformation of the intermediate goods, which is an extremely different thing. In the real world, the tangible and visible role of fixed capital is not to transfer its value to the products, or to deserve the payment of a profit rate, but to help increase the productivity of the main actor of the production process, which is human labor.

Since the whole treatment of fixed capital by Sraffa is based on the particular question of the age of machines, as we shall see below, it is also important to note that in the real world there is no such thing as a once and for all clearly defined age of a machine, which would remain the same during its whole participation in the production process. In *Production of commodities by means of commodities*, we are clearly at a technical level, but even if we stay at this level there is no such thing as a predefined lifespan or age of whatever machine or piece of equipment, since this age will depend on many factors, like the intensity of the use of the machine.

A machine working eight hours a day with a single shift obviously will not have the same lifespan as exactly the same one working 24 hours a day with several shifts. The quality of maintenance, which can vary from a firm to another due to multiple factors, as well as during its own lifetime, can also greatly alter the real life duration of an equipment. Furthermore most machines are not even used during the whole duration of their nominal lifetime, for the well-known reason that they quickly become obsolete. New and cheaper or more ‘productive’ machines are indeed produced each year and after a few years make production with older machines become no longer competitive. Hence the replacement of machines becomes
indispensable, even a long time before the day when they would have been worn out. The result is that in the real economic world the composition of a collection of machines and the way they are used vary continuously.

From a theoretical point of view all this is all the more annoying that the whole treatment of fixed capital by Sraffa, in § 76 of his book, consists in establishing a sub-system based on as many equations as there are separate processes which correspond to the successive ages of a given machine. He thus uses a method first developed by Torrens, who had introduced the notion of “residue of capital” in his “Essay on the Production of Wealth” (Torrens, 1821, pp. 28-29). Sraffa also indicates that “the quantities of means of production, of labor and of the main product are equal in the several processes in accordance, with the assumption of constant efficiency during the life of the machine” (Sraffa, 1960, p.78). This is hardly compatible with the fact that neither the lifespan nor the efficiency of a machine can ever be determined at any given point of time.

Nevertheless this is not the main criticism that can be made of the treatment of capital by Sraffa. To consider it, let us recall that for each machine, there is a sub-system having as many equations as the successive ages of this machine, for age 0, 1, 2,...,n, where n is the lifetime of each machine. Each of the n equations represents the joint production of good G and of a machine of age 1 to n (on the right-hand side) by a machine of age 0 to n - 1 (on the left-hand side). This sub-system covers a whole range of years, as many as the lifespan of the machine. With a proper treatment, Sraffa then removes n - 1 equations corresponding to the machines of intermediate ages to finally obtain a single equation containing only the newly-produced machine, of age 0. This equation is the following (see § 76 of the book):

$$M_0 p_{m0} \frac{r(1+r)^n}{(1+r)^n - 1} + (A_g p_a + \dots + K_g p_k)(1+r) + L_g w = G_g p_g$$

The first term represents the annual depreciation of the machine, i.e. the value supposed to be transferred by the machine to the final good G for a given year n.

However, when we come back to a whole system of production for a single given year, we therefore turn, as Sraffa does in § 83:

> “from the standpoint of the life-progress of a single machine to the stand point of a complete range of n similar machines each being one year older than the preceding one, and thus forming a group such as we might find in a self-replacing system. The requirement that the life-sum of the depreciation quotas should be constant and independent of the rate of profits is now embodied in the fact that under all circumstances such a group is maintained simply by bringing in a new machine each year” (Sraffa, 1960, § 83, p.83).

All this is quite coherent and means that in this self-replacing production system, we have n machines in operation (from age 0 to age n - 1), with on the left-hand side of the n equations like the one above n different depreciations. As Sraffa clearly demonstrates also, and this
demonstration is right, the price of the machines of the intermediate ages can vary with the rate of profits, but for a given rate, as it appears from Figure 6 (in § 83), the sum of these different prices is always equal to the initial value of the machine $p_{m0}$.

Up to now, everything might seem perfectly correct, but a problem arises when we realize that, when we come to the calculation of production prices, we are no longer in the sub-system in which it was innocuous to make appear $n$ different machines of age 0 to $n-1$ (the machine of age $n$ being withdrawn from the production process). Indeed, in a self-replacing state there is one and only one machine, of age 0, which is produced in a given year, with its own equation representing the conditions of its production. In this equation the quantity of the machine $M_0$ of age 0 (since it is new) appears on the right-hand side. And for a good G, and/or any other good in the production process of which this machine is used, including its own, there are in total as many equations for each good as the number of years $n$ corresponding to the lifespan of this machine $M_0$.

Turning now to the way this whole system works, on the left-hand side of each of these equations there is a value:

$$M_0 p_{m0} \frac{r(1+r)^n}{(1+r)^n - 1}$$

which represents the contribution to production or the “transfer of value” of these machines of age 0 to $n-1$ to the price of the goods that they help produce. We also know that the sum of these depreciations is $M_0 p_{m0}$. But we are now in the real system itself, where only new machines of age 0 are really produced each year, each of these different machines with its own equation of production, and no longer in the sub-system where there was joint production of the machines of various ages. This implies that there are no equations corresponding to the real production of machines from age 1 to $n$, because these machines have in fact been produced previously, in earlier periods. This means that on the right-hand side of all these equations these machines of age 1 to $n$ cannot and do not appear.

The interesting thing is that, as a consequence of this situation, and for the whole production system, when we sum up all the equations in which a machine $M_0$ appears, we have on the left hand-side a value $M_0 p_{m0}$ corresponding to the sum of the depreciations, and on the right hand side exactly the same value $M_0 p_{m0}$ of the newly produced machine $M_0$. This signifies clearly that, whatever the lifespan of the machines, in a self-replacing state the value of the quantity of the machines which is produced in each period corresponds exactly to the quantity which is supposed to “disappear” in the production process, where this value is supposed to be transferred to the value of the goods that they contribute to produce, to the tune of the total depreciation affecting the same machines of various ages.

The inescapable conclusion of this analysis is that there is no such thing as a surplus for machine $M$, nor for any machine, since the demonstration performed so far can obviously be
generalized. Although all this can easily be understood intuitively (since in a self-replacing
state there is no net production of fixed capital, but only its replacement), we have therefore
analytically established a very important result, which is that in Sraffa's system as long as
fixed capital is supposed to transfer its value to the goods produced, *there is no surplus of
fixed capital*. This means reciprocally that, supposing that there is a surplus, it cannot include
any fixed capital good.

**5. The contradictions of Sraffa's system with fixed capital**

At the point where we have arrived, we cannot but observe that we face a double
contradiction: indeed Sraffa defines production as a circular process (meaning that what
comes out of the process also enters or rather re-enters into it) through which a surplus is
created.

However, what is to some extent circular in the production process, as exposed in *Production of commodities...*, is the production of intermediate goods and fixed capital, this last one being assimilated to a particular type of intermediate good with a span of life longer than the production period, because both are supposed to enter into and come out of the same production process. This explains why they appear on both sides of the equations describing this process. But at the same time, in a self-replacing state, where the whole Sraffa system is located, and as we have already shown, there is no such thing as a surplus of either intermediate goods or fixed capital, because fixed capital is itself a kind of intermediate good, and therefore there should be no production according to Sraffa's own definition of this concept.

As for consumption goods, we have already shown that these goods do not enter into the
production process, but are destroyed (either instantaneously or more or less quickly, if they
are durable) in the consumption process. It is clear therefore that there is no circular
production process concerning consumption goods. However, since these consumption goods
cannot be found on the left-hand side of the equations describing the process, but appear
necessarily on the right-hand side, the difference between both sides, for consumption goods
only, is necessarily made of the whole production of these consumption goods. One might
possibly say that it constitutes a "surplus", although it would be an artificial one, because for
the process as a whole the quantities of intermediate goods should normally be deducted from
this "surplus", in order to obtain the true surplus of the system. But this is obviously
impossible, because of the heterogeneity of the goods on both sides of the equations.

This brings us to a paradox: it is the Standard net product which has to be divided between
wages and profits, but because what might actually be called a surplus and constitutes this net
product is made only of consumption goods, then these profits should be devoted entirely to
buying consumption goods! More importantly, this obviously contradicts the statement made
by Sraffa according to which "the rate of profits in the Standard system thus appears as a ratio
between quantities of commodities irrespective of their price” (Sraffa, 1960, bottom of p. 24).

Indeed there is no such thing as a ratio between consumption goods and intermediate goods!

Another contradiction comes from the fact that Sraffa’s system is built in such a way that:

> "...the ratio of the net product to the means of production would remain the same whatever variation occurred in the division of the net product between wages and profits and whatever the consequent price changes" (Sraffa, 1960, § 28, p. 24).

However, even with all other parameters remaining unchanged, if the rate of profit changes from one period to another during one of the $n$ periods corresponding to the lifespan of a machine, the depreciations in the value of this machine already registered during the preceding periods will naturally remain unchanged, but the amount of the depreciations that will take place in the following periods, after the change in the rate of profit, will also change necessarily. It follows that the sum of the $n$ depreciations for a given machine will no longer be equal to the value of the machine, but will be lower or higher.

The same phenomenon would take place each time that the actual lifespan of a given machine would become shorter or longer than the original or nominal one. In both cases, either because of a change in the rate of profit or in the real life span of a machine, the resulting change in the overall amount of the depreciations will change the proportions in which the machine enters in the production process, and therefore the nature of the Standard system and the whole price system! This is indeed not compatible with Sraffa’s system.

Before going further, let us go back to the definition of fixed capital as a kind of “long life” variety of intermediate goods, since we can now better realize that this vision is incorrect. Indeed what appeared on the left hand side of the equations, and that we did not questioned, has to be revisited, since it is clear that it was not the machine itself, but a purely virtual element, i.e. its depreciation, which varies, as Sraffa explains, both with the age of the machine and with the rate of profit. The only thing which does not vary, as we already pointed out, is the sum of the depreciations for the $n$ machines of age 0 to $n-1$ in operation for a given period, which is always equal to $p_{m0}$, but only for a given rate of profit. This virtual quantity introduces an irreducible element of heterogeneity with the other intermediate goods: although they disappear in the process in their initial form, but only to be transformed, intermediate goods enter into it as real goods, and not as virtual ones.

# 6. The consequences for the Standard system

## 6.1. The notion of basic goods

An important conclusion that we can draw from these observations is that most goods are not basic goods, whose main property, as defined by Sraffa in § 6 of *Production of commodities...* is that they “...enter (whether directly or indirectly), in the production of all commodities”. Indeed, since these goods are also produced, it means that one has to find them both on the
left and right-hand side of the equations defining the system of production. We have shown that consumption goods do not meet this criterion, and at this stage we must admit that it is also the case for fixed capital goods. Indeed it is not these goods themselves, but only their depreciation that enters into the production process, and this depreciation, as defined by Sraffa, is not a tangible element, which could be put on the left-hand side of the equations in a non-contradictory manner. From this we must derive that fixed capital goods have to be considered as final goods, which like consumption goods can be found on the right-hand side, but not on the left-hand side of the equations.

In fact this corresponds to the treatment applied to depreciation by Keynes, who devotes the whole appendix of Chapter 6 of the *General Theory of Employment, Interest and Money* to the question of what he calls “the user cost”. He defines the user cost “...as the reduction in the value of the equipment due to using it as compared with not using it” (Keynes, 1936, p. 66), and indicates that aggregate income, equivalent to aggregate supply price, is equal to $A – U$, or “as being net of aggregate user cost”. It is clear that this user cost is quite similar to what Sraffa calls “the annual charge to be paid for interest and depreciation” for a machine, and that as we have seen, unlike Keynes, he includes it wrongly in the production cost.

As a result, the only goods that can be found on both sides of the equations are clearly the intermediate goods, strictly defined as goods that disappear (in the sense only of being transformed) within the production process, where they can be used in the production of any other good (final or intermediate) as well as in their own production (directly or indirectly). They are thus the only goods that can be called basics in the sense that Sraffa gives to this word. There is however no reason why there should be any surplus of these intermediate goods, which are produced for each of them in the same total quantity as the sum of the quantities used in all the various industries, as Sraffa calls them.

However, as Sraffa himself shows it, the fact that only intermediate goods *can be* basics does not mean that all intermediate goods *are* actually basic goods. As soon as § 6 of his book, Sraffa indicates indeed rightly that apart from luxury goods per se, there are two kind of intermediate goods that do not enter in the production of *all* commodities, and therefore are not basics: these are luxury goods that enter only in their own production, or only in the production of other luxury goods. Things turn out to be a little more complicated in part II of the book (“Multiple-Products Industries and Fixed-Capital”), where joint production is introduced, before being used to deal with fixed capital.

In chapter VIII, “the Standard System with Joint Products”, of “*Production of Commodities by Means of Commodities*”, Sraffa demonstrates that there is also a Standard system and a Standard commodity in case of joint production, even though the Standard commodity can include negative quantities for some individual commodities. He indicates also that there are still three types of non-basics:

1. “Products which do not enter the means of production of any industries” (...).
2. "Products each of which enters only in its own means of production" (...).

3. "Products which only enter the means of production of an interconnected group of non-basics" (Sraffa, 1960, §57, p. 57).

Category 1) corresponds for Sraffa to what he calls luxury goods, but that we prefer to call final goods, for reasons which should be now easily understandable. For the two other categories, and in order to reformulate this description in a somewhat simplified way, we must consider that if there are intermediate goods that do not enter in the production of other intermediate goods, and enter only in the production of final goods (consumption goods or fixed capital good), and that we could name for this reason "ultimate intermediate goods", then they cannot be basic goods, and as such cannot be part of the Standard commodity. This leads to recognize that all intermediate goods cannot be basic goods, that in other words there is no identity between the set of intermediate goods and the set of basic goods of a given economic system, and to further restrict the content of the Standard commodity.

## 6.2. The notion of surplus

To be sure, some particular goods might prima facie be considered as basic goods, with an amount produced even greater than the amount used as an input (ignoring the differences already mentioned between their nature as an input and as an output). This is the case of some agricultural products.

But, as we have already shown in the second chapter about Ricardo's conception of production (see above p. 33), it appears so because there is a biological mechanism of organic production which as such does not come obviously from spontaneous generation, but precisely from the transformation of elementary goods, free or not. Indeed, using Lavoisier's formula, who has no reason to be false in economy, "nothing is lost, nothing is created, everything is transformed." These elementary goods are the oxygen and carbon in the air, Hydrogen and oxygen from the rain, nitrogen, potassium and phosphorus in soils, among others, which thanks to the biological mechanism of photosynthesis (among other processes at work), allow to harvest some agricultural products in higher quantities than those which have been sown. But the difference in quantities is used in the production of other goods, and in terms of material balance, the overall process is balanced, and it cannot logically be otherwise.

The same thing is true for production in its economic sense. Even for agricultural products, apart from the part that is self-consumed by small individual producers, products sold on the market for final consumption are not formally those harvested, because they have passed through various successive stages of preparation, packaging and transportation, which make them different from those goods which have been harvested, so that for them the concept of the surplus as a difference is not appropriate. As for mineral raw materials, mineral deposits already exist underground prior to their extraction, and it is clear that there can no more be any surplus in the form of any directly consumable good for any ore whatsoever.
These observations have an important consequence on the very nature of what can be called a
surplus. They do not prevent us from considering that production is a circular process, at least
for the goods that are part of the rightly-named “circulating capital”. In their case, one could
even imagine that there is a surplus, but that would only be true for the subsystem, necessarily
incomplete, which produces the circulating capital with circulating capital, since only a
portion of the circulating capital is used in its own production process. But this surplus is a
purely artificial one, it is an artefact, deriving from a mathematical property of this sub-
system.

Indeed, at the level of the production system as a whole, all the circulating capital is used: the portion that is not directly used for the production of the circulating capital itself obviously enters in the production of final goods: consumer goods and fixed capital. And this circulating capital is always fully utilized: even in the case of expanded reproduction, the circulating capital produced in excess during a period compared to that produced during the previous period is fully used in the increased production of various goods, be they intermediate or final, during this same period. Indeed production prices should be considered actually as reproduction prices, which concern only a single period, during which there cannot be any surplus of intermediate goods (assuming the absence of stockpiles).

As for final goods, since they do not enter as such in the production process, they exist only as
an output of the system, and not as an input. Therefore, it is true that we might think of a
difference at an individual level between the output and the input: since for each of them this
input is zero, the total quantity produced might be considered as a kind of individual surplus.
But if we did so we would forget to deduct all the intermediate goods that are on the left-hand
side of the equation of the system, and that have actually entered, directly or indirectly, into
the production process of this final good. Similarly, at the level of all of these final goods, this
is not true either, because at this global level all of the intermediate goods which enter as
inputs into the whole production process have to be deducted from the whole output, which is
impossible to perform due to the heterogeneity of both types of goods. As a consequence, a
so-called surplus cannot be quantified, and has therefore no logical meaning.

**6.3. The notion of Standard commodity**

We already explained in the last chapter how Sraffa builds up his Standard commodity, as an
invariable Standard of prices. Let us recall that Sraffa devotes Chapter IV of his book to this
Standard commodity, which he defines as a commodity that would not itself change in value
when the distribution between wages and profit changes. He notes in § 24 that the perfect
composite commodity that could play this role:

> "...is one which consists of the same commodities (combined in the same proportions) as does the aggregate of its own means of production." (Sraffa, 1960, § 24, p. 21).
In § 26 he calls the set of equations taken in the proportions of the Standard commodity, the
Standard system. He continues by saying that:

> "...in any actual economic system there is embedded a miniature system which
can be brought to light by chipping off the unwanted parts" (Sraffa, 1960, §26,
p. 22).

and he adds that: "...this applies as much to a system which is not in a self-replacing state as
to one which is". He then: "...takes as unit of the Standard commodity the quantity of it that
would form the net product of a Standard system employing the whole annual labor of the
actual system", calling it "the Standard net product."

Finally in § 28 Sraffa defines as the Standard ratio:

> "...the rate by which the total product of the Standard system exceeds its
aggregate means of production, or the ratio of the net product to the means of
production of the system" (Sraffa, 1960, § 28, p. 23), underlining that:

> "... the possibility of speaking of a ratio between two collections of
miscellaneous commodities without need of reducing them to a common
measure of price arises of course from the circumstance that both collections
are made up in the same proportions – from their being in fact quantities of the
same composite commodities." (Sraffa, 1960, § 28, p. 23-24).

It is easy to see that all this demonstration can be carried out only because the very first
assumptions made by Sraffa establish an *ad hoc* classification of the goods, where all the
goods, apart from the luxury goods as he names them, can in effect play both roles of means
of production and of final products, which entitles them to being basic goods (except for some
particular kinds of intermediate goods which are non-basics). This is what leads him to his
particular treatment of fixed capital goods. However, we have shown that this classification is
wrong, and that the only goods which are simultaneously means of production and products of
the system are the intermediate goods, as part of the circulating capital, but that there cannot
be a surplus of them, because they are (all of them) transformed in the process of production
of final goods, be they consumption goods or machines, i.e. fixed capital goods.

To be sure, and we cannot deny it, at a theoretical level it is possible to extract from an actual
economic system a Standard system, with its Standard commodity, Standard net product and
Standard ratio. But it is a theoretical artefact: the Standard ratio R, for instance, is nothing
more than a kind of efficiency indicator or index reflecting the degree or intensity according
to which basic commodities are used in their own production, knowing that the remaining
basics are not a surplus, but are necessarily used, directly or indirectly, in the production of all
the other commodities. Moreover, if we suppose that wages are set as zero, R is then the
maximum share of profits when they equal the whole surplus, i.e. the Standard net product.
But because it corresponds to a surplus of this particular part of intermediate goods that is
made of basics, R cannot be identified to profits, which are supposed to represent at least the
fixed capital part of final goods.

It follows therefore that in an actual economic system where no confusion is made between
the different types of goods, it is impossible to define, at the scale of the whole economic
system, either a Standard commodity such as Sraffa's, or a Standard system, or a Standard net
product made of true final goods, or a Standard ratio. Moreover Sraffa tells us in § 43 that:

> "...the last remaining use of the Standard net product is as a medium in terms of which the wage is expressed – and in this case there seems to be no way of replacing it" (Sraffa, 1960, §43, p. 38).

Consequently the fact that there is no such thing as a Standard net product made of true final goods implies in particular that the wage cannot be expressed in terms of this medium. Therefore we cannot but conclude on this point that the famous relation of proportionality $r = R(1-w)$ between the wage and the rate of profits established by Sraffa, and at the heart of his theory, cannot reflect the true functioning of an actual economic system, where for instance the share of profits should be such as to allow for the purchase of fixed capital goods.

# 7. The intractable problem of land and natural resources

An additional problem arises with the introduction of non-produced means of production, such as land and mines, in Sraffa's system, because this makes it reach again its conceptual limits. Indeed, in the same way that goods appearing only as products but not as means of production – such as consumer goods or fixed capital goods – i.e. in fact all final goods, are not basics, and cannot be part of the standard commodity, conversely land, or more specifically in this case the different land qualities, are among the means of production used to produce agricultural products, but obviously are not part of the product. They are not basic goods, but their existence implies the payment to their owners of a rent which is part of the production price.

When an agricultural product, such as wheat, is produced by several lands of different qualities (fertilities), to determine a price system including the price of this product (wheat), implies that the rent be removed from the system of equations, which would include otherwise more unknowns than equations. This in turn implies that the equation used for the production of wheat in the system of equations is that which corresponds to the land without rent, i.e. the least "fertile" land, but which yields however the average rate of profits. The rent of other more "fertile" lands, with a lower production cost per unit, can then be obtained as a differential rent. It is also clear that the least fertile land is one for which the production cost per unit of output is the highest, or conversely that which produces the lowest amount for a given production cost.

The problem, long known (and recognized by Sraffa himself), is that the determination of the least fertile land depends on the cost of production, i.e. on the price system. Fertility is not an
absolute, exogenously given and intrinsic quality of the land, but a relative parameter, which itself depends on the price system. The introduction of land and generally of non-produced means of production and of their income, that constitutes rent, drives the system into a circular reasoning: to determine the price system, we must know which one of the different lands is the land without rent, i.e. the least "fertile" one, which implies to know the price system.

At a second level, the price system varies depending on the variation of the rate of profit or of the wage level. As a result, for different levels of the rate of profit and wages, we necessarily get different price systems, which leads to changes in the production cost of all goods produced with non-produced means of production, such as land or natural resources. These modifications in the price system in turn change the order of land fertility or of the 'productivity' of these natural resources.

Again, we see that "fertility" cannot be defined as a parameter which would be independent of distribution. The equation defining the method of production for the "marginal" land or natural resource can vary with distribution, with different equations corresponding to different marginal lands, depending on these changes in distribution. Therefore the system of equations defining the methods of production when using the least fertile land or other non-produced 'marginal' resources does not remain the same when distribution changes, which implies that the corresponding Standard commodity also varies. Therefore, since the rate of profit or the level of wages are defined in terms of the Standard commodity, it is the very notion of a variation in this rate of profit or level of wages that becomes irrelevant, because it is by definition impossible to compare two different Standard commodities: it is clear that each Standard commodity consists of heterogeneous goods in different proportions, each set corresponding to a different system of equations.

It must therefore be acknowledged that Sraffa's theory fails to provide a coherent conceptual framework which would be able to deal in a non-contradictory or non-circular manner with non-produced means of production and rent, in a way that would maintain the internal consistency of this theory.

**8. Conclusions on Sraffa's system and the Standard commodity**

Sraffa's system is a brilliant construction, which had the great merit to dismantle the neoclassical theory of distribution for which wages and profits are determined by the marginal productivity of labor and capital at the equilibrium. It shows well that when multiple heterogeneous goods and production are introduced, wages and profits have nothing to do with any "marginal productivity", but have to be defined as a share of a net product. It nevertheless results from the above demonstration that the system faces insurmountable problems, in fact partially recognized by Sraffa himself, especially when fixed capital and non-produced means of production are introduced in the system.
Moreover, as soon as a price system is needed to express and determine a state of distribution, and to the extent that the price system itself reflects a state of distribution, we fall into a circular reasoning: we must know the price system to measure distribution, but one must know the distribution to determine this system. Sraffa resolves this dilemma by defining an invariable Standard of value, which is invariable with respect to distribution, and in which distribution can therefore be expressed independently of prices. This invariable Standard is the Standard commodity, which leads to the well-known formula $r = R(1-w)$. But this is achieved by paying a heavy theoretical price:

* The Standard commodity is in fact a composite aggregate of heterogeneous goods, which are by definition basic goods, thus appearing to the right and left of the equations giving the prices. This Standard commodity made of basic goods allows to some extent to reason, with all the limitations which have been reported, as if we were in the universe with a single good of the neoclassical theory or the Essay on Profits of Ricardo, of which ‘Production of commodities…’ constitutes an attempt of generalization. But whereas prices in this Ricardo’s system had the dimension of a quantity of wheat, in the theory of production prices all prices have the dimension of the Standard commodity, which is at the root of the other problems encountered by the theory;

* Any change in the composition of the Standard commodity is a change of system which is unintelligible, since it is impossible to compare two different Standard commodities, i.e. two sets of different heterogeneous goods, with thus two different dimensions. Each Standard commodity corresponds to a period during which the methods of production cannot change : in the real world this implies that such a period must be extremely short;

* Since production is generally a continuous process, with a continuous change in the methods of production and therefore in the Standard commodity, this prevents from considering this Standard of value as a unit of measure for money, which has to link the present to the future. This makes it impossible to introduce into this system a money with Keynesian characteristics, meaning – as we already saw in chapter 1, that its elements are pure numbers without any dimension, or scalars. I had shown that Keynes’s units of measurement were not compatible with a Standard of value made of heterogeneous goods in a paper included in a collective book published in 1975: “Controversies on the Keynesian System”.

* Therefore a change in distribution becomes unintelligible as soon as the methods of production change by the slightest amount, since the composition of the Standard commodity changes simultaneously;

* While prices are expected to ensure the reproduction of the system, no mechanism is provided to connect the expense of wages and profits to this reproduction, unless we
assume that workers like capitalists share the same set of basic goods (that are in fact a
particular kind of intermediate goods) in the same proportions, which is absurd;

* One could even say that the system cannot be strictly speaking the capitalist system, since the difference between workers and capitalists is suppressed by the fact that what they share is in fact the same Standard commodity.

Let us conclude that on the basis of the above analysis commodities do not produce
commodities. As we shall see in more details in the following chapters, labor, helped by
capital as a catalyst, produces commodities. Maybe Sraffa was a little aware of that, which
would help explain the enigmatic character of his subtitle to "Production of commodities...":
"Prelude to a critique of economic theory".

References

Flamant, Christian (1975). *Remarques sur le chapitre 4 de la Théorie générale: le choix des unités de mesure* (Remarks on chapter 4 of the General Theory: the choice of units). In *Controverses sur le Système Keynésien* (*Controversies on the Keynesian System*). Paris: Economica.

Sraffa, Piero (1960). *Production of commodities by means of commodities. Prelude to a critique of economic theory*. Cambridge (UK): Cambridge University Press.

Torrens, Robert (1821). *An Essay on the Production of Wealth*. London: Longman, Hurst, Rees, Orme and Brown. University of California Digital library [online]. Available at: https://archive.org/details/essayonproductio00torrrich
Accessed 25 Aug 2018.
# PART II - PRODUCTION AND VALUES

"The laws and conditions of the Production of wealth partake of the character of physical truths. There is nothing optional or arbitrary in them. Whatever mankind produce, must be produced in the modes, and under the conditions, imposed by the constitution of external things, and by the inherent properties of their own bodily and mental structure...It is not so with the Distribution of wealth. That is a matter of human institution solely. The things once there, mankind, individually or collectively, can do with them as they like. They can place them at the disposal of whomsoever they please, and on whatever terms. Further, in the social state, in every state except total solitude, any disposal whatever of them can only take place by the consent of society, or rather of those who dispose of its active force". John Stuart Mill, in Principles of Political Economy (1848), Book II, Chapter I, § 1.
# Chapter 5. Production and values

This chapter will introduce the concept of production as a transformation process, considered as the physical transformation of intermediate goods into final goods. We will show that this transformation comes from the intervention of labor considered as an activity, and that on the other hand, fixed capital as a commodity is not transformed in this process, and is not an activity. Its intervention consists therefore in increasing the productivity of human labor. This is the reason why human labor alone can be at the origin of value, and why fixed capital cannot transfer its value to the product. Values must therefore be defined at the level of production, and human labor can be a standard of value first because it is not a commodity, and secondly because through its payment in money per unit of time it can be reduced to abstract labor, and acquire as such a dimension, which is the dimension of time.

## 1. The analytical distinction between production and distribution

The previous chapters showed that Ricardo as well as Sraffa were unable to build up a coherent theory of production and prices. Ricardo wanted to resolve the question of distribution and its variations within the framework of his theory of production, but failed because of the impossibility of constructing an invariable standard. Sraffa succeeded to some extent in building up this standard, but he had to abandon the idea of determining distribution within his system, because he had to introduce the profit rate from outside, with wages still determined within the system as being the remaining share of a surplus. However this implied a number of theoretical flaws, as was explained in the previous chapter. It will be shown now that there is a way to overcome these difficulties, which necessitates to go further than Sraffa in delinking production and distribution, and to adopt another conception of production. This will allow to solve in a coherent way the problem of determination of values and prices.

Maybe the first economist to separate clearly production from distribution was John Stuart Mill, to whom we refer in the quotation from his “Principles of Political Economy”, published in 1848, and highlighted above at the beginning of part II of this book. Although this distinction is contrary to the whole neo-classical tradition and paradigm for which incomes come from the productivity of production factors, it should go nevertheless without saying since it is indeed both a logical and chronological requirement. For commodities to be distributed among individuals (be they workers or capitalists, this does not matter), they must exist first, which means logically that they must have been produced first. In turn this implies that production must have taken place in chronological time before distribution.

This point, which corresponds to our second basic principle, is so straightforward that there is no need for justifying it through lengthy explanations. Suffice it to say that various modes of production differ in the social organization of production, but that production as a physical and technical process of creation of goods remains basically the same whatever the type of social organization.
To give but a simple example supporting this assertion, let us remark that the way water mills function did not changed when the mode of production based on slavery of the Roman Empire was replaced by the feudal mode of production, or later on by the capitalist mode of production. However nobody will deny that the social rules governing the distribution of the product of these mills, for instance wheat ground by them to be turned into flour, were deeply modified.

As Mill had well understood, the distinction between production and distribution is therefore a methodological requirement. It is to conform to this unescapable requirement that we will deal first with production, and try to deal at this sole level with the question of values.

# 2. Production as a transformation process

## 2.1. The background

It has been established so far that the problems encountered by Sraffa's theory could not but come from its very roots, i.e. from his initial assumption that production is a circular process: this leads him to an erroneous definition and classification of the goods which are part of this process, either as entering into it or as coming out of it. While Sraffa rightly demonstrates that there is no such thing as a "productivity" of capital, this is also what leads him to consider fixed capital goods as if they were intermediate goods, and to put the 'depreciation' of capital on the left of his equations, which implies that fixed capital transfers its value to the products, an assumption rejected by Keynes.

Does this mean that production must therefore be regarded, as Sraffa also wrote, to characterize the alternative view, as "a one-way avenue that leads from factors of production to consumption goods" (Sraffa, 1960, Appendix D, p. 111)? The previous chapter seemed not entirely to validate this view either. It has been shown indeed that production is a transformation process only for intermediate goods, and that part of these goods are used collectively in their own production, which indeed can be considered to some extent as a circular process.

Even in such a process, which concerns in fact all of the primary commodities, and however circular it may appear, the apparent existence of a surplus comes from a kind of "optical effect". Indeed in the production of agricultural products, we do not "see" this kind of particular intermediate goods which are brought about by nature: oxygen taken from the air, water coming from the rain, nitrogen obtained from the soil, etc. In other words, all the components which have been transformed into agricultural products do not appear as inputs on the left side of the production equations, but they do not appear on the right side either as outputs, because they have not been economically produced. Therefore they cannot be part of a surplus. Similarly, the mineral products extracted from underground already exist, and as such are not physically "produced": they are only taken from the geological layers where they have been enclosed for millions of years. But this must nevertheless be considered as a production in its economic sense, because it involves a transformation, if only of location.
At the individual level of each primary commodity, none of them directly enters indeed in its own production: no crude oil is directly used in the production of crude oil, or no apple in the production of apples: these are the apple seeds which are sowed to give apple trees...These are the phenomena which seem to give rise at the individual level of each good to a physical surplus, because it appears as a product on the right hand side of its own production equation, without appearing on the left hand side of the same equation, whereas there is in fact no such thing at the scale of the whole economic system. At this level indeed every intermediate commodity which appears as an output on the right side of the equations describing the production process appears in identical quantities on the left side.

To be sure, a good part of intermediate goods does not enter totally in the production of other intermediate goods, but it is because the remaining part is transformed into final goods, which implies that globally there cannot be any surplus of intermediate goods. As for final goods, they are either consumption goods or fixed capital goods, and for this very reason never enter again to be transformed in the production process, once this transformation has occurred and is over. There is no longer any circularity here, and we stand truly within the framework of a one way avenue, from primary commodities and intermediate goods to final goods, i.e. within a conception of production as a pure transformation process. This is what Sraffa missed to understand, because for him all goods, and even fixed capital ones, are intermediate goods, with the only exception of luxury goods.

If we reduce a final good to all of the components assembled to make it, and these components to their own components, and so on up to the most elementary components, there is no doubt that we will go back to a finite number of primary commodities or raw materials, all of them being from natural origin, and coming from the soil or from underground. These basic elements have indeed all been produced by cultivation or harvesting, in the case of agricultural products, or have been mined and extracted from the ground, as far as minerals are concerned. As we pointed out above, some may have come from the air, like water from rain or various gaseous elements. This confirms that production is primarily a natural and physical process, even though it is simultaneously a technical one.

Another observation which is easy to make is that the production of primary commodities has become over the centuries a more and more complex and capitalistic activity: apart from fixed capital, it requires also more and more intermediate goods, as well as energy, which formally speaking is not a material good, but appears in any production process. Among others, Steve Keen in his blog in the Real World Economic Review, pointed out recently the paramount role played by energy into production: "The abiding weakness of all schools of economics, ever since the Classicals - including today's Neoclassical and Post Keynesian schools, which are normally at pains to point out how superior one is to the other - is this failure to acknowledge the key role of energy in production" (Keen, September 5, 2016). Keen even goes as far as putting energy at the same level as labor or capital, in a production function derived from the Cobb-Douglas one.
Although the important role of energy is certainly undeniable, one must nevertheless underscore that there is a big difference between labor and energy, which is that energy is itself produced, whereas labor is not a good which would be produced: labor is not on the right side of the production equations. Moreover, as soon as we abandon the idea of identifying labor to a quantity of wage goods (and we showed why this conception is wrong), labor cannot be replaced by any good, be it energy, in these equations.

This leads us to a new basic principle, which is the following:

**Principle 11:** Production, or the activity which creates goods, is a transformation process, which creates new goods through the transformation of pre-existing or already produced goods. As such production does not create any physical surplus.

We know also that the human activity consisting in the production of goods is labor, and we are thus led to go back to and endorse Adam Smith's view put forward in the first sentence of the introduction to the "Wealth of Nations", and according to which: "The annual labor of every nation is the fund which originally supplies it with all the necessaries and conveniences of life which it annually consumes, and which consist always either in the immediate produce of that labor, or in what is purchased with that produce from other nations." (Smith, 1776, volume I, p. 1).

Let us recall finally that it was also shown in chapter 4 that fixed capital, while it plays an indispensable role in the production process by helping to transform intermediate goods, is not itself transformed in this process: no part of any machine, be it a specific part or a fraction of the whole machine, has ever been incorporated into one of the final goods in the production of which it is involved. Fixed capital is not either a factor of production in the sense that it would have a measurable "productivity", independent of distribution and of the price system.

We can thus formulate our next principle, with its corollary:

**Principle 12:** Labor is the human activity which produces goods through the transformation of other goods, and as such is the only factor of production.

All the above observations therefore explain why capitalist production can only be defined as a labor process of transformation of intermediate goods into final goods, which takes place in the context of specific social relations, where wages are paid in money to workers who do not own fixed capital goods. Fixed capital plays an important role in this process, but as a catalyst which as such is present and physically unchanged from the beginning up to the end of the production process, and increases in considerable proportions the productivity of labor. Therefore we can add to principle 12, as a corollary:

**Corollary to Principle 12:** Fixed capital, as well as energy, is not an activity, but a produced good, and as such is not a factor of production. As a good its specificity is that it is not transformed in the production process, but intervenes into it through increasing the productivity of labor. Fixed capital goods are therefore final goods.
On the basis of these premises, how can we progress in the understanding of the production
process? If we adopt a step by step approach, we can start by focusing first on its physical
properties, before going one step further in addressing the production process as a social
process.

**2.2. Production as a physical process**

Let us consider an economy with a number $s$ of different and heterogeneous goods. These goods are designated by index $i$, and the branches producing them by index $j$. Quantities produced or outputs are designated by $A_j$, with $i=1,...,s$, and $j=1,...,s$. From what we learned from the analysis of Sraffa's system and the preceding observations, we can classify these $s$ goods in three main categories:

- The first one, with indices going from 1 to $k$, comprises intermediate goods which enter directly or indirectly into the production of all the other commodities. We will call them intermediate goods of the first type. This category includes all of the primary commodities, and also all the goods that enter into the production of these primary commodities, as well as the goods entering in the production of the latter, and so on. As we previously pointed out these goods in their initial form do not enter usually in their own production: for instance raw coal does not enter as such in the production of raw coal, or coarse wheat in the production of wheat (it has to be transformed first into a different good, i.e. wheat seeds). Moreover, and although it is not always the case, a part of these goods may enter directly in the production of final goods, and the remaining part not: for instance most agricultural food products are not consumed as harvested, because they need some processing, packaging and transportation before they can be consumed. But they still enter indirectly in the production of final goods. Therefore, and for the sake of simplicity, we will not consider these sub-categories, which at this stage are not theoretically important.

- The second category, with indices going from $k+1$ to $n$, comprises intermediate goods which do not enter into the production of the previous category, but enter only in the production of final goods, and can also enter in their own production. They are generally appearing downstream in the production process. We will call them intermediate goods of the second type.

- The third category, with indices going from $n+1$ to $s$, is made of final goods, which never enter in the production process, because they have reached their final form, and will not be transformed further. This category includes the two sub-categories of consumption goods and fixed capital goods.

We now designate by index $i$ the branch of origin whose outputs are the means of production, i.e. the intermediate goods used as inputs in the production of goods $j$; and by $A_{ij}$ the quantity of good $i$ used for the production of good $j$. This means that the methods and techniques of production are given. They are based on science and technology, which is to a
large extent independent of the economic system, and as such are external to the object of economic theory, although the choice of techniques among several available ones can obviously be an economic problem, which is as such relevant for economic theory and certainly needs to be addressed by it.

It must also be noted that at a given point in time, several different techniques can co-exist in a branch producing a particular good, or even in the case of a single producer, if a firm has for instance several different factories producing this same good. This implies that the theoretical level where we stand is neither a pure micro-economic level, nor a macro-economic one, but rather what we could name a "meso-economic" level. This implies also that all the quantities of the various goods used in the production of a given good correspond to the total quantities used in a given branch to product this given good, and that the average proportions in which they are combined may not correspond to the actual proportions of any specific individual producer of this given good.

We also introduce labor, designating by $L_i$, with $i=1,...,s$, the "quantities of labor" used in the production of commodity $i$. The question of the nature of a "quantity of labor" is going to be addressed below, but here too it must be emphasized that these quantities are total quantities.

Then the production system can be represented as follows, being understood that this scheme is not a matrix, and that in particular the indices $i$ and $j$ do not correspond to the usual matrix notation of rows and columns:

$$ egin{array}{lcccccccccc} A_{11} & \dots & A_{k1} & 0 & 0 & 0 & \text{and} & L_1 & \rightarrow & A_1 \\ \dots & \dots & \dots & 0 & 0 & 0 & \dots & \dots & \dots & \dots \\ A_{1k} & \dots & A_{kk} & 0 & 0 & 0 & \text{and} & L_k & \rightarrow & A_k \\ A_{1,k+1} & \dots & A_{k,k+1} & A_{k+1,k+1} & \dots & A_{n,k+1} & \text{and} & L_{k+1} & \rightarrow & A_{k+1} \\ \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots \\ A_{1n} & \dots & A_{kn} & A_{k+1,n} & \dots & A_{nn} & \text{and} & L_n & \rightarrow & A_n \\ A_{1,n+1} & \dots & A_{k,n+1} & A_{k+1,n+1} & \dots & A_{n,n+1} & \text{and} & L_{n+1} & \rightarrow & A_{n+1} \\ \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots \\ A_{1s} & \dots & A_{ks} & A_{k+1,s} & \dots & A_{ns} & \text{and} & L_s & \rightarrow & A_s \end{array} $$

We must stress once more that fixed capital goods (the "machines") only appear on the right side of this representation, where they constitute a part of the goods numbered from $n + 1$ to $s$. This is so because they are obviously produced and are an output of the system, but do not enter in this process to be transformed and therefore do not disappear in their original form during its implementation. They are present in the same condition at the beginning and at the end of the process, with the only difference of being older, by a time which represents the duration of this process. The situation is not the same, however, for the goods used for their maintenance, which are transformed, one way or another (they can even disappear), in the course of the production process.
It must also be stressed again that the above scheme is not a matrix, because the quantities $A_{ij}$ are not scalar but are quantities $A_{ij}$ of different heterogeneous goods with various dimensions. Because of this heterogeneity they cannot be added, which explains the absence of the addition symbol +, and at this stage we did not use either this sign + to note the intervention of labor in this transformation process.

## 2.3. Homogeneity and the nature of abstract labor

In order for the heterogeneous goods which are transformed and produced through the production process that we just described to be thought of as homogeneous commodities, we need to find a common property which can be assigned to them, and by definition this property will be named their value, in the sense that the existence of this common property makes them equivalent in terms of itself. What comes out of the above scheme is that the only common property of all these goods is to have been produced by labor, and this is precisely this property which had lead the first classical economists as well as Marx to develop the labor theory of value.

However, as we already indicated (see chapter 1), labor itself corresponds to a number of heterogeneous activities, with various skills differing broadly from one branch to another, and within a given branch the existence of various levels of skills, generally linked to various levels of education and responsibility. Marx himself was quite conscious of the problem, and addressed it by developing the concept of abstract labor, intended to overcome all these differences that result in the obvious heterogeneity of labor.

Marx introduces this concept of abstract labor in section 2 “The twofold character of the labor embodied in commodities”, of chapter 1 of “Capital”. There he first defines useful labor in relation to use-values: “in the use-value of each commodity there is contained useful labor, i.e., productive activity of a definite kind and exercised with a definite aim” (Marx, 1887, p. 31). A few lines further he adds: “So far therefore as labor is a creator of use-value, is useful labor, it is a necessary condition, independent of all forms of society, for the existence of the human race; it is an eternal nature-imposed necessity, without which there can be no material exchanges between man and Nature, and therefore no life” (Marx, 1887, p. 31). And finally he arrives to abstract labor by indicating: “But the value of a commodity represents human labor in the abstract, the expenditure of human labor in general” (Marx, 1887, p. 32).

The developments that Marx devotes to this question are finally summarized in the very last paragraph of this section 2 of chapter 1 of “Capital”, where he writes: “On the one hand all labor is, speaking physiologically, an expenditure of human labor-power, and in its character of identical abstract human labor, it creates and forms the value of commodities. On the other hand, all labor is the expenditure of human labor-power in a special form and with a definite aim, and in this, its character of concrete useful labor, it produces use-values” (Marx, 1887, p. 33).
On the basis of this section and of such citations, some commentators have interpreted abstract labor as being physiological labor, but it is not very helpful in a theory seeking a common dimension in order to measure values, because physiologically labor is an activity which differs from an individual to the other, each of them having his own physiology, with its own strength, etc.

Most commentators, however, have interpreted abstract labor sociologically, such as the great Hungarian philosopher and sociologist Georg Lukacs, a friend and associate to Max Weber. In his well-known book "History and Class Consciousness", he writes: "If we follow the path taken by labor in its development from the handicraft via co-operation and manufacture to machine industry we can see a continuous trend towards greater rationalization, the progressive elimination of the qualitative, human and individual attributes of the worker. On the one hand, the process of labor is progressively broken down into abstract, rational, specialized operations so that the worker loses contact with the finished product and his work is reduced to the mechanical repetition of a specialized set of actions. On the other hand, the period of time necessary for work to be accomplished (which forms the basis of rational calculation) is converted, as mechanization and rationalization are intensified, from a merely empirical average figure to an objectively calculable work-stint that confronts the worker as a fixed and established reality" (Lukacs, 1920, p. 88). To be sure, from a sociological point of view, this Marxist concept of abstract labor is certainly quite relevant, but as such, and within the framework of our quest for an economically relevant concept, it offers no particular basis for quantification, because abstract labor as defined has no particular dimension.

Nevertheless, Marx himself, although he certainly did not master the concept of dimension, was quite conscious of this necessity of reasoning in terms of quantities. This appears quite clearly if we refer to one of his works anterior to the publication in 1867 of his main book "Capital". This previous work is a second and rather long unpublished draft of chapter 6 of the later, dating back to 1864. Let us quote a particularly meaningful passage of this draft chapter: "Just as the owner of commodities is only interested in the use value of the commodity as the vehicle of its exchange value, so the capitalist is only interested in the labor process as the vehicle and instrument of the valorization process. Within the production process too - in so far as it is a valorization process - the means of production continue to be simply monetary values, for which the particular material shape, the particular use value in which this exchange value is expressed is a matter of indifference, just as labor itself does not count within the production process as productive activity of a particular useful character, but as the substance that creates value, as social labor in general which is being objectified, and of which the only interesting aspect is its quantity" (Marx, 1864, E-book).

To go further towards the understanding of abstract labor, one must in fact refer to a long forgotten economist, Isaak Illich Rubin. In his book "Essays on Marx's Theory of Value", originally published in 1924, he devotes one full chapter to discuss this particular concept of abstract labor, and shows well that the answer to the question of homogeneity can be found in Marx himself, despite all the complexities of his writings. Rubin emphasizes the fact that "Marx considers the quantity of working time expended by the worker the basic property
which characterizes the quantitative determination of labor" (Rubin, 1924, chapter 14, p. 156).
He refers in particular to Marx's book, "A Contribution to the Critique of Political Economy",
published in 1859, and which is a kind of first draft of "Capital". He points out that "when we
consider the distribution of total social labor among individuals and branches of production
(...) different quantities of labor appear as different quantities of labor-time. Thus Marx
frequently replaces labor with labor-time, and examines labor-time as the substance
materialized in the product (Critique, pp. 23, 26)" (Rubin, 1924, p. 156).

Although Rubin, like Marx, is certainly wrong in identifying labor-time with "a substance materialized", he is completely right in highlighting time as the quantitative factor which transforms labor in abstract labor by stressing its dimension as labor-time, i.e. as time itself, because time, as a well-established physical concept, does not appear to have particular varieties. If we recognize that, we have therefore the solution to the homogeneity problem, because labor, despite of the many dimensions it can have as a physiological, social or productive activity, has only one dimension, which is not a substance materialized, and which is common to all its possible kinds and varieties, and it is time as a physical dimension, meaning a dimension of the physical and material world.

**2.4. Values have the dimension of time**

From what we just understood, it derives that various types of this particular activity which
labor is have no common dimension other than the time during which any particular type of
activity is performed. Therefore the only way for abstract labor to be valid as a scientific
concept underlying the concept of value is to be considered as labor reduced to its sole
dimension of time, being aware that time as such is indeed a dimension of the physical world:
time is a dimension of physics and one of the seven fundamental physical quantities in both
the International System of Units and the International System of Quantities.

To acknowledge that is to understand that values are magnitudes that have the dimension of
time, which is a perfectly homogeneous dimension, and that the values of commodities must
be expressed in units of time. Moreover time presents the interest to be defined as a physical
entity by the International Systems in charge of this task, and which also define the unit of
time. It is equally important to note that this dimension of commodities, i.e. their value as
time, can be defined at the stage of their production, seen as a purely physical and technical
process, before they enter into exchanges and are the object of any distribution.

At this stage we can therefore formulate a new and fundamental principle:

**Principle 13:** The only way to think of commodities as homogeneous through a common property, which is by definition their value, is to consider that they are produced by abstract labor, i.e. labor reduced to its only dimension of physical time. Therefore values, and commodities through this common property, have the dimension of time.
**Corollary to Principle 13:** Since time in which values are expressed is the time of physics, measured in physical units like hours, days, weeks, months or years, values at the level of production of commodities must be considered as "physical values".

**2.5. Fixed capital cannot transfer its value to the product**

Although we already explained in sub-section 2.1. why fixed capital was not a factor of production, the last principle explains also why fixed capital cannot transfer its value to the products that it helps to produce through its catalytic role. Indeed from the last principle it derives that value is not a substance, but a property of commodities (among a number of other properties), which itself has the dimension of time. Each commodity has therefore the characteristic of having a value with the dimension of a time, and this characteristic is intrinsic to each of them.

It must be stressed that such a property, which as such is not a substance and has no material content, is not separable from the commodity to which it corresponds, and as such does not exist independently of this commodity. Since we know that fixed capital itself is not an intermediate commodity, and as such does not enter into the production process and is not transformed in this process, it follows that its value cannot be transferred either.

This gives us the following additional principle:

**Principle 14:** Since fixed capital is not itself transformed in the production process, its value as an intrinsic characteristic of commodities cannot be transmitted to the products in the production of which fixed capital participates as a catalyst.

A somewhat paradoxical consequence of this principle is that the intervention of fixed capital, through its increase in labor productivity, usually reduces the value of the products.

As for intermediate commodities, we have the corresponding corollary:

**Corollary to Principle 14:** The value of intermediate commodities is transferred to a product because these commodities themselves are transferred to this product through their transformation as a part or an element of it, in the course of its production process. In some cases (like energy) this transformation can be a disappearance.

This being said it remains to be seen whether values can be concretely determined, a question which is going to be addressed in the next chapter.

**References**

Keen, Steve (2016). Incorporating energy into production functions. Real World Economic Review Blog, September 5, 2016.

https://rwer.wordpress.com/2016/09/05/incorporating-energy-into-production-functions/ Accessed 6 May 2018.
Lukacs, Georg (1920). Reification and the Consciousness of the Proletariat. In *History and Class Consciousness - Studies in Marxist dialectics* (pp. 83-222). New Edition 1967. Cambridge, Massachusetts, USA: the MIT Press.

Marx, Karl (1867) *Capital: A Critique of Political Economy. Volume I - Book One: The Process of Production of Capital*. Source: First English Edition of 1887. Marx/Engels Archive [online]. Available at: https://www.marxists.org/archive/marx/works/download/pdf/Capital-Volume-I.pdf Accessed 8 Jul 2018.

Mill, John Stuart (1848). *Principles of Political Economy with some of their Applications to Social Philosophy*. London: Longmans, Green and Co.

Rubin, Isaak Illich (1924). *Essays on Marx's Theory of Value*. English edition: Rubin, I. (1973). Montreal, (Canada): Black Rose Books.

Smith, Adam (1776) An Inquiry into the Nature and Causes of the Wealth of Nations. Published by Political Economy [online]. Available at: http://political-economy.com/wealth-of-nations-adam-smith/ Accessed 25 Aug 2018.
# Chapter 6. A first determination of values as physical values

At the stage where we now stand the question of distribution of the product is not yet a relevant one, including its distribution among various categories of workers performing various kinds of labor or more or less skilled labor, which is therefore not taken into account. We can thus consider that labor can and in fact must be represented through its dimension of time considered only as chronological time, without any consideration for the differences between various categories of labor. Now that we have linked values and time, and since time is a homogeneous and quantifiable dimension of the physical world, we can represent production in a mathematical way allowing for the determination of values, which will be calculated as quantities of labor time.

To this end, let us start first with a series of assumptions underlying the calculation of these values, which will therefore be pure physical values.

## 1. The assumptions

At the outset, let us indicate that we will from now on deal with objects that are made homogeneous by the establishment of a common property, their value, which has the dimension of time measured in physical time units: hours, or days, or weeks, of months, or years. It is the reason why we will no more speak of goods, which are heterogeneous, but of commodities. This being said, the assumptions behind this determination of values are the following:

1) There is only one method of production for each commodity, which is in fact the average method of production for the branch producing a particular commodity;

2) Each branch produces only one commodity, which avoids the problems of joint production (although these problems can be solved, which will be done later, because taking them into consideration now would introduce unnecessary complexity);

3) The only primary factor of production is labor, which is represented by the time during which it is performed, and as such is abstract labor devoid of all of its other possible dimensions. There is fixed capital (it is part of the product, as a particular type of final good), but it is not a factor of production in the usual sense;

4) The period of production is arbitrary, which may result in the fact that the production of some final goods which takes a particularly long time to be produced is not fully completed during a period (they can be registered as unfinished commodities);

5) All commodities have this same period of production, which therefore is not a purely technical data;
6) Although inputs are produced first during the period and outputs are produced when the necessary inputs are available, production is a continuous process, and therefore we will consider that all inputs and all outputs are produced during the same period.

In the sake of simplicity, the whole system is normalized by taking as a unit the overall produced quantity of each commodity.

Then we note $a_{ij}$ the ratio $\frac{A_{ij}}{A_i}$, with $\sum_{j=1}^{s} a_{ij} = 1$, meaning that, as we previously showed, there is no surplus of intermediate goods.

$a_{ij}$ is thus the share of total production or output of branch $i$ ($A_i$) used as an input in the production of commodity $j$ by branch $j$ ($A_{ij}$). They are called technical coefficients, being understood that these coefficients are not defined like in National Accounting (i.e. $\frac{A_{ij}}{A_j}$).

$l_i$ is the labor-time it takes in branch $i$ to produce one unit of commodity $i$, in time units.

On these bases, Morishima demonstrated that there are two different methods for the determination of values, in a book published in 1973: “Marx’s Economics : a Dual Theory of Value and Growth”. These two methods will now be developed, one after the other.

# 2. A first method for the determination of values

## 2.1. A first definition of values

On the above bases, let us first consider that there are $s$ commodities, and that their values are the elements of a vector of values $V = [v_1 \ ... \ v_s]$. Since it is a row vector these elements are named $v_j$ (with $j = 1, ..., s$). Let us define the value $v_j$ of one unit of commodity $j$ as the overall labor time it takes to produce commodity $j$.

Since commodity $j$ has been produced through the transformation of various other intermediate commodities, $v_j$ can be defined first as the sum of the direct labor time $l_j$ needed to produce commodity $j$, plus the indirect time corresponding to the value of the commodities that have been transformed to produce commodity $j$, which is the value of its means of production, i.e. the intermediate commodities $i$ used in its production, be they of the first type, with $i = 1, ..., k$, or of the second type, with $i = k + 1, ..., n$.

$$ \text{Thus } v_j = a_{1j}v_1 + a_{2j}v_2 + \dots + a_{jj}v_j + \dots + a_{nj}v_n + l_j \quad \text{or} \quad v_j = \sum_{i=1}^{n} a_{ij}v_i + l_j \quad (1) $$

In passing it must be stressed that $a_{jj}v_j$ is always equal to zero. Indeed in supposing that it would not be the case, then we could write:

$$ v_j - a_{jj}v_j = a_{1j}v_1 + a_{2j}v_2 + \dots + 0 + \dots + a_{nj}v_n + l_j \quad (2) $$
And then we could change the dimension of the equation by dividing both members by $1-a_{jj}$ which in any case proves that $a_{jj}=0$ for any $i=j$, or that no commodity can enter directly in its own production, or in other words can be transformed into itself.

## 2.2. The calculation of values with the first method

On the basis of our first definition, we have therefore one equation for the determination of the value of each commodity, which means that there are $s$ equations for $s$ commodities. And since $a_{ij}$ coefficients are scalars (i.e. pure numbers) and all of the values have the same dimension as the $l_j$, which is time measured in time units, for instance one hour, or one month, or one year, the production system which has been exposed above (see on page 102) can now be made homogeneous. It is therefore possible to represent this production system mathematically, on the basis of the above equation extended to all commodities. In order to represent this system in matrix format we can try to use a square matrix which could be inverted, like in Sraffa's production system. But such a matrix taking into account all the $s$ goods would be quite strange, since with its $s$ rows and $s$ columns it would appear as follows:

$$ \begin{bmatrix} v_1 & \dots & v_s \end{bmatrix} \begin{bmatrix} a_{11} & \dots & a_{1k} & a_{1,k+1} & \dots & a_{1n} & a_{1,n+1} & \dots & a_{1s} \\ \vdots & \ddots & \vdots & \vdots & \ddots & \vdots & \vdots & \ddots & \vdots \\ a_{k1} & \dots & a_{kk} & a_{k,k+1} & \dots & a_{kn} & a_{k,n+1} & \dots & a_{ks} \\ 0 & \dots & 0 & a_{k+1,k+1} & \dots & a_{k+1,n} & a_{k+1,n+1} & \dots & a_{k+1,s} \\ \vdots & \ddots & \vdots & \vdots & \ddots & \vdots & \vdots & \ddots & \vdots \\ 0 & \dots & 0 & a_{n,k+1} & \dots & a_{nn} & a_{n,n+1} & \dots & a_{ns} \\ 0 & \dots & 0 & 0 & \dots & 0 & 0 & \dots & 0 \\ \vdots & \ddots & \vdots & \vdots & \ddots & \vdots & \vdots & \ddots & \vdots \\ 0 & \dots & 0 & 0 & \dots & 0 & 0 & \dots & 0 \end{bmatrix} + \begin{bmatrix} l_1 & \dots & l_s \end{bmatrix} = \begin{bmatrix} v_1 & \dots & v_s \end{bmatrix} $$

In matrix notation this might be written as the following system: $VA+L=V$

However it is obvious that this notation leads us nowhere, because such a system cannot be solved. Indeed, despite the fact that matrix $A$ is a square matrix, with its $s$ rows and $s$ columns, it is not indecomposable, because inter alia all of the elements of the last $n+1$ to $s$ rows are zeros.

To understand how we could resolve the system, let us first use a simplified presentation with submatrices. The initial system can thus be transcribed into the following one:

$$ \begin{bmatrix} V_1 & V_2 & V_3 \end{bmatrix} \begin{bmatrix} A_{11} & A_{12} & A_{13} \\ 0_{21} & A_{22} & A_{23} \\ 0_{31} & 0_{32} & 0_{33} \end{bmatrix} + \begin{bmatrix} L_1 & L_2 & L_3 \end{bmatrix} = \begin{bmatrix} V_1 & V_2 & V_3 \end{bmatrix}, \quad \text{which we name system A} $$
The significance of the various submatrices and vectors appearing in this system deserves a few explanations. Let us begin with the row vector of values $V = [v_1 \dots v_s]$: it can be decomposed into $V_1$, the vector of values of intermediate commodities of the first type, $V_2$ the vector of values of intermediate commodities of the second type, and $V_3$ the vector of values of final commodities:

$$V_1 = [v_1 \dots v_k], V_2 = [v_{k+1} \dots v_n], V_3 = [v_{n+1} \dots v_s]$$

As for the different submatrices with positive elements $A_{11}$, and $A_{12}$ to $A_{23}$:

- Submatrix $A_{11}$ is a square matrix of dimension $k$, its elements reflect the fact that the first $k$ intermediate goods enter collectively in their own production: they are primary commodities or commodities entering directly or indirectly in their production, and we now call them intermediate commodities of the first type. In passing, as we saw, at individual level no such commodity enters in its own production, which means that $a_{ij} = 0$ for every $i = j$ (this constitutes a difference with the situation of wage-goods in the Ricardian system);

- Submatrices $A_{12}$ and $A_{13}$ are rectangular matrices of dimensions $(k, n-k)$ and $(k, s-n)$ respectively. Their elements reflect the fact that these same $k$ intermediate commodities of the first type enter into the production of what we now call intermediate commodities of the second type (this corresponds to submatrix $A_{12}$ and generally (but not always) into the production of final commodities (this corresponds to submatrix $A_{13}$));

- Submatrices $A_{22}$ and $A_{23}$ are a square matrix of dimensions $(n-k, n-k)$ and a rectangular matrix of dimensions $(n-k, s-n)$ respectively. Their elements reflect the fact that the $n-k$ intermediate commodities of the second type enter into their own production (this corresponds to submatrix $A_{22}$) and generally (but not always) into the production of final commodities (this corresponds to submatrix $A_{23}$).

Similarly, the significance of the various $0$ submatrices is the following:

- Submatrix $O_{21}$, of dimension $(n-k, k)$, reflects the fact that the production of the $k$ intermediate commodities of the first type does not require any input from the $n-k$ branches producing intermediate goods of the second type, because these last ones do not enter directly or indirectly in the production of the previous intermediate goods, and therefore in the production of all goods, but only enter in their own production and in the production of final goods;

- Submatrices $O_{31}$ of dimension $(s-n, k)$, $O_{32}$ of dimension $(s-n, n-k)$ and $O_{33}$ of dimension $(s-n, s-n)$ reflect the fact that the $s-n$ final commodities by definition do not provide any inputs to the $k$ branches producing intermediate goods of the first type (this corresponds to submatrix $O_{31}$), neither any inputs to the $n-k$ branches producing
intermediate goods of the second type (this corresponds to submatrix $O_{32}$), nor any inputs to their own production as final commodities (this corresponds to submatrix $O_{33}$).

As for the row vectors of labor quantities, each of them corresponds to one of the three types of commodities:

- $L_1 = (l_1, ..., l_k)$ is the row vector of the quantities of direct labor used in the production of commodities $A_1$ to $A_k$, respectively, measured in physical time and in time units;

- $L_2 = (l_{k+1}, ..., l_n)$ is the row vector of the quantities of direct labor used in the production of commodities $A_{k+1}$ to $A_n$, respectively;

- $L_3 = (l_{n+1}, ..., l_s)$ is the row vector of quantities of direct labor used in the production of commodities $A_{n+1}$ to $A_s$, respectively.

As noted, system A cannot be resolved, because its matrix is not indecomposable, and has to be reduced. In order to do so, we can first try to suppress the three zero submatrices at the bottom of matrix A, called submatrices $O_{13}$, $O_{23}$ and $O_{33}$. This gives us a second system with a first decomposition of matrix A, now transformed in matrix A', and which can be represented as follows:

$$[V_1 \ V_2 \ V_3] \begin{bmatrix} A_{11} & A_{12} & A_{13} \\ 0 & A_{22} & A_{23} \end{bmatrix} + [L_1 \ L_2 \ L_3] = [V_1 \ V_2 \ V_3], \quad \text{which we name system A'}$$

But this system cannot be resolved either. Indeed matrix A' is not a square matrix, but a rectangular matrix of dimension $(n, s)$, which again is not indecomposable, because of the presence of a submatrix $O_{21}$ on the lower left side of matrix A'.

In order to simplify this system one might think of merging submatrices $A_{12}$ and $A_{13}$ into one single submatrix $A_2$, and submatrices $A_{22}$ and $A_{23}$ into one single submatrix $A_3$. This would imply merging vectors $V_2$ and $V_3$ into one vector $V_2$, and vectors $L_2$ and $L_3$ into one single vector $L_2$. We would then obtain the following system:

$$[V_1 \ V_2] \begin{bmatrix} A_1 & A_2 \\ 0 & A_3 \end{bmatrix} + [L_1 \ L_2] = [V_1 \ V_2]$$

One cannot but be struck by the resemblance of this system with the Ricardian system of chapter 2 (p. 45). By analogy to the latter, and in matrix format, we could therefore try similarly to write this system by splitting it in two sub-systems, in the following way:

$$V_1 = V_1 A_1 + L_1 \quad (3)$$

$$V_2 = V_1 A_2 + V_2 A_3 + L_2 \quad (4)$$
However it is obvious that matrix equation (2) above cannot be written because the expression $V_2A_3$ displays a big problem of dimensionality: indeed vector $V_2$ has $s-k$ elements corresponding to all the final commodities ($s-n$), plus the intermediate commodities of the second type ($n-k$), whereas submatrix $A_3$ has only $n-k$ rows, corresponding only to these last commodities. Multiplying $V_2$ by $A_3$ is therefore totally impossible.

The failure of this simplification attempt shows once more that it is impossible to assimilate wage-goods in the Ricardian sense, where they are viewed as intermediate goods, and final goods, these last one being absent on the left side of the equations.

We must therefore revert to system A', and in order to resolve it, rewrite it in a way which takes into account the existence of the three different types of goods introduced in sub-section 2.2. of the present chapter: intermediate goods of the first type, of the second type, and final goods. The system has therefore to be split into three subsystems, which allows to represent it in the following matrix format, where we have three matrix equations:

$$V_1 = V_1 A_{11} + L_1 \qquad (5)$$

$$V_2 = V_1 A_{12} + V_2 A_{22} + L_2 \qquad (6)$$

$$V_3 = V_1 A_{13} + V_2 A_{23} + L_3 \qquad (7)$$

Only the first one is an independent matrix equation, from which we get:

$$V_1 = L_1 (I - A_{11})^{-1} \qquad (8)$$

Then, knowing $V_1$, the vector of values of intermediate commodities of the first type, from equation (8) we get $V_2$, the vector of values of intermediate commodities of the second type:

$$V_2 = (I - A_{22})^{-1} (V_1 A_{12} + L_2) \qquad (9)$$

And knowing $V_1$ and $V_2$, from equations (8) and (9) we obtain $V_3$, the vector of values of final commodities, without any other additional calculation than what appears in equation (7):

$$V_3 = V_1 A_{13} + V_2 A_{23} + L_3$$

In passing let us underline that it would not be relevant in this framework to consider the $k$ intermediate commodities of the first type as basics in the sense of Sraffa, because what distinguishes basics is not only the fact that they enter directly or indirectly in the production of all the other commodities, but also, as we showed it, the fact that a profit rate has to be levied on them. However there is no profit in this analysis, which remains strictly at the stage of production, i.e. before any distribution has taken place.
### 3. A second method for the determination of values

#### 3.1. A second definition of values

The second method is based on a different way of defining values, which can also be found in Marx<fnref n="5" />. The value $v_j$ of a commodity $j$ is now defined as the sum of the totality of direct labor time used to produce one unit of it. This includes the direct labor time used in its own production, i.e. in the transformation of the intermediate commodities used in this final process, plus the direct labor time used to produce these same intermediate commodities, and in turn the direct labor time used to produce their own means of production, and so on.

Since this time is proportional to the quantities of each intermediate good that enters ultimately, directly or indirectly, into the production of all the other intermediate or final goods, to make the calculation implies that we allocate this direct labor time to the quantity of each commodity which enters, directly or indirectly, in the production of all of the other ones.

Let us call $x_{ij}^*$ the quantity of commodity $i$ used to produce one unit of commodity $j$. Then let $x_{ij}$ be the share of the total production of commodity $i$ which is used directly and indirectly to produce one unit of commodity $j$. Thus we have:

$$x_{ij} = \frac{x_{ij}^*}{x_i}, \text{ where } x_{ij} \text{ is a scalar} \qquad (10)$$

Using the new definition of this second method, the values $v_j$ that we want to determine for the first $k$ intermediate commodities are defined as:

$v_j = x_{1j}l_1 + x_{2j}l_2 + \dots + x_{kj}l_k$ (we recall that there are no $x_{ij}$ for $i > k$, i.e. for final commodities)

$$\text{Or } v_i = \sum_{j=1}^{k} l_j x_{ji} \qquad (11)$$

#### 3.2. The calculation of values with the second method

From our new definition, it is clear that we must first determine the quantities $x_{ij}$ appearing in the last equations. In order to do so without unnecessarily complicating the demonstration, we will assume that there are no intermediate commodities of the second type, and that therefore we have only $n$ commodities, i.e. $k$ intermediate commodities of the first type and $n-k$ final commodities.

Then, with the first method, the system of three matrix equations (1), (2) and (3) above for determining values becomes a system with only two matrix equations:

<footnote n="5">⁵ This is well explained by Morishima (op. cit.) where chapter 1 provides a demonstration.</footnote>
$$V_1 = V_1 A_{11} + L_1 \qquad (12)$$

$$V_2 = V_1 A_{12} + L_2 \qquad (13)$$

Which gives us, as solutions to these equations:

$$V_1 = L_1 (I - A_{11})^{-1} \qquad (14)$$

$$V_2 = L_1 (I - A_{11})^{-1} A_{12} + L_2 \qquad (15)$$

Going back now to the second method, let us call $X_i$, with $i = 1,...,n$, the column vector giving the quantities of the $n$ commodities that are produced by the system:

$$X_i = \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ \vdots \\ x_n \end{bmatrix}, \text{ or } X_i = \begin{bmatrix} X_1 \\ - \\ X_2 \end{bmatrix}, \text{ with } X_1 = \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_k \end{bmatrix} \text{ and } X_2 = \begin{bmatrix} x_{k+1} \\ x_{k+2} \\ \vdots \\ x_n \end{bmatrix}$$

Like with the first method for calculating values, and in order to make this calculation possible, we need first to transform some of the quantities that we are dealing with into scalars, which is done by taking each of them as a unit.

Let us call $L_i$ the row vector of the direct quantities of labor time used in the production of the $n$ commodities $X_i$:

$L_i = l_1, ..., l_k, l_{k+1}, ..., l_n$, which in vector terms can be written:

$$L_i = [L_1 | L_2], \text{ with } L_1 = l_1, l_2, ..., l_k, \text{ and } L_2 = l_{k+1}, l_{k+2}, ..., l_n$$

When we calculated the values of commodities with the first method, the quantities of each intermediate commodity that enter *directly* in the production of the other ones were known, since knowing them comes down to knowing the methods of production, which are among the data. These methods of production are given by the $a_{ij}$ coefficients, which as we recalled are named technical coefficients in standard input-output analysis. These $a_{ij}$ coefficients were indeed defined as the quantities of intermediate commodities $i$ (or outputs of branch $i$) used directly as inputs in the production of a commodity $j$ (output), with $i = 1, ..., k$, and $j = 1, ..., n$.

In order to calculate these same values with the second method, we need now to go one step further and determine the quantities $x_{ij}$, which include the inputs of commodities $i$ entering
indirectly in the production of commodities j, through their use in the production of other direct inputs, and so forth.

If we try to represent the system globally, as we did for the first method, the resolution of the system would therefore consist in the determination of all the elements of a matrix X which would look as follows:

$$ \begin{bmatrix} x_{11} & \dots & x_{1k} & x_{1,k+1} & \dots & x_{1n} \\ \vdots & \ddots & \vdots & \vdots & \ddots & \vdots \\ x_{k1} & \dots & x_{kk} & x_{k,k+1} & \dots & x_{kn} \end{bmatrix} = [X_{11} \ | \ X_{12}] \text{ of dimension } k \times n $$

Since each $x_{ij}$ represents the total amount of direct and indirect intermediate commodities $i$ required as inputs to produce one unit of commodity $j$, in order to determine this amount, we can write for each intermediate commodity $i$ a series of equations using the fact that technical coefficients $a_{ij}$ are a series of already known data (the methods of production), which are themselves elements of the following matrix:

$$ \begin{bmatrix} a_{11} & \dots & a_{1k} & a_{1,k+1} & \dots & a_{1n} \\ \vdots & \ddots & \vdots & \vdots & \ddots & \vdots \\ a_{k1} & \dots & a_{kk} & a_{k,k+1} & \dots & a_{kn} \end{bmatrix} = [A_{11} \ | \ A_{12}] \text{ of similar dimensions } k \times n $$

It is obvious that it is impossible to resolve a system using these matrices as such, because both are rectangular matrices and therefore are not invertible. It is indeed impossible to multiply matrix $A$ by matrix $X$, because $A$ has more columns than $X$ has rows, although it is possible to multiply matrix $A_{11}$, which has only $k$ columns, by matrix $X$, which has $k$ rows.

Therefore, if we limit ourselves first to the $k$ intermediate commodities of the first type, we can multiply matrix $A_{11}$ by matrix $X_{11}$ (the first $k$ columns of matrix $X$). We also know that matrix $A_{11}$ is productive, because vector $X_1$ (with $i = 1, ..., k$), which gives the total production of the first $k$ intermediate commodities, is non negative and such that $X_i \ge A_{1i}X_i \ge 0$. We are indeed in an open Leontief model, and this productivity entails some implications, which have been exposed among alia by Peterson and Olinick, in an article on Leontief models and Markov chains which they published in 1982 in "Mathematical modelling". This implies first that:

$$ \sum_{j=1}^{k} a_{ij} X_j \le X_i \text{ for } i = 1, 2, ..., k \qquad (16) $$

This equation means that all the production of the first $k$ intermediate commodities is not used in their own production, since a part of this production is used in the production of the $n-k$ final commodities. This also implies that:
$$ \sum_j \left( \sum_i a_{ij} \right) X_j = \sum_i \sum_j a_{ij} X_j < \sum_i X_i \qquad (17) $$

Therefore, since all intermediate commodities of the first type enter by definition in the production of all commodities other than themselves (i.e. in the production of final commodities), for any intermediate commodity $i$ we must have:

$$ \sum_j a_{ij} < 1 \qquad (18) $$

Inequality (16) also implies (by subtracting $a_{ii}X_i$ from both members), that:

$$ 0 \le \sum_{j \ne i} a_{ij} X_j < X_i - a_{ii} X_i = (1 - a_{ii}) X_i \qquad (19) $$

Therefore $a_{ii} < 1$ for every $i$ and matrix $I - A_{11}$ is non singular. On this basis a theorem guarantees that sub-stochastic matrix $A_{11}$ is productive. Then matrix $I - A_{11}$ is invertible and $(I - A_{11})^{-1} > 0$, with a solution which is the production vector $X_1 = (I - A_{11})^{-1} F$ (with $F$ as the vector of final demand).

Going back to this production vector $X_1$, i.e. the column vector giving the quantities of the $k$ intermediate commodities of the first type that are produced by the system, and because matrix $X_{11}$ is also productive, we must write, for each of these $k$ commodities, an equation which indicates that it is the sum of all its total uses, direct and indirect, entering as its outputs in the production of all the other $k$ commodities, to which must be added the share of its output entering in the production of final commodities.

In an open Leontief model, this additional share is usually considered as final demand and for this reason it is generally called $f_i$ for commodity $i$. However it must be understood that it is not a demand for final commodities themselves, but only for the intermediate commodities entering in the production of these final commodities. Then we can write:

$$ x_1 = x_{11} + \dots + x_{1j} + x_{1k} + f_1 $$

$$ x_i = x_{i1} + \dots + x_{ij} + x_{ik} + f_i $$

$$ x_k = x_{k1} + \dots + x_{kj} + x_{kk} + f_k $$

To continue in matrix notation, let us have:
$$
X_1 = \begin{bmatrix} x_1 \\ \vdots \\ x_k \end{bmatrix} \qquad A_{11} = \begin{bmatrix} a_{11} & \cdots & a_{1k} \\ \vdots & \ddots & \vdots \\ a_{k1} & \cdots & a_{kk} \end{bmatrix} \qquad X_{11} = \begin{bmatrix} x_{11} & \cdots & x_{1k} \\ \vdots & \ddots & \vdots \\ x_{k1} & \cdots & x_{kk} \end{bmatrix} \qquad F_1 = \begin{bmatrix} f_1 \\ \vdots \\ f_k \end{bmatrix} \quad \text{and} \quad I_i = \begin{bmatrix} 1 \\ \vdots \\ 1 \end{bmatrix}
$$

Where $I_1$ is used to represent what is known as the summation vector, i.e. a column vector of 1's, of appropriate dimension (here $k$). Then in matrix notation, Miller and Blair explained in their book, published in 2009: "Input-Output Analysis: Foundations and Extensions", that these equations that we just wrote above can be summarized as:

$$X_1 = A_{11}X_{11}I_1 + F_1 \quad (20)$$

It is well known that such an equation represents an open Leontief model, and can be easily resolved in the sense that the production vector $X_1$ as well as matrix $X_{11}$ can be determined for any given vector of “final demand” $F_1$. Let us show it by assuming that this final demand is represented by one unit for each of the $k$ intermediate commodities. Then the above equations become:

$$x_1 = x_{11} + \dots + x_{1j} + x_{1k} + 1$$

$$x_i = x_{i1} + \dots + x_{ij} + x_{ik} + 1$$

$$x_k = x_{k1} + \dots + x_{kj} + x_{kk} + 1$$

The $a_{ij}$ seem to have disappeared – and thus matrix $A_{11}$, but it is just because the elements $x_{ij}$ of these equations are obtained through a multiplication by matrix $A_{11}$. In order to determine the elements $x_{ij}$ appearing above, and before writing the detailed equations giving each of these elements, an example will help us to understand how they are obtained.

Let us take for instance element $x_{li}$, i.e. the total quantity of commodity 1 (as a share of its total product) entering in the production of commodity $i$. It is the sum first of the share of commodity 1 going into its own production $a_{11}x_{li}$ (which should be 0, because as we saw $a_{11}$ is normally equal to 0), plus the share of the production of commodity 1 going in the production of the share of the production of commodity 2 which goes itself in the production of commodity $i$, i.e. $a_{12}x_{2i}$ ..., and so on, up to the share of the production of commodity 1 going in the production of the share of the production of commodity $k$ which goes itself in the production of commodity 1, i.e. $a_{lk}x_{lk}$. On top of these intermediate uses of commodity 1, we must also take into account the “final demand” for commodity 1, which we have assumed to be equal to 1. Therefore the corresponding equations are:
$$x_{11} = a_{11}x_{11} + \dots + a_{1k}x_{k1} + 1$$

$$x_{21} = a_{21}x_{11} + \dots + a_{2k}x_{k1}$$

$$x_{i1} = a_{i1}x_{11} + \dots + a_{ik}x_{k1}$$

$$x_{k1} = a_{k1}x_{11} + \dots + a_{kk}x_{k1}$$

$$x_{1i} = a_{1i}x_{1i} + \dots + a_{1k}x_{ki}$$

$$x_{2i} = a_{2i}x_{1i} + \dots + a_{2k}x_{ki}$$

$$x_{ii} = a_{i1}x_{1i} + \dots + a_{ik}x_{ki} + 1$$

$$x_{ki} = a_{k1}x_{1i} + \dots + a_{kk}x_{ki}$$

$$x_{1k} = a_{11}x_{1k} + \dots + a_{1k}x_{kk}$$

$$x_{2k} = a_{21}x_{1k} + \dots + a_{2k}x_{kk}$$

$$x_{ik} = a_{i1}x_{1k} + \dots + a_{ik}x_{kk}$$

$$x_{kk} = a_{k1}x_{1k} + \dots + a_{kk}x_{kk} + 1$$

In matrix notation this system becomes, first for the full system of equations:

$$X_{11} = A_{1}X_{11} + I$$

$$\Rightarrow X_{11} - A_{11}X_{11} = I$$

$$\Rightarrow X_{11} = (I - A_{11})^{-1} \qquad (21)$$

We recognize in this equation (21) the well-known solution to open Leontief models.

Whereas for a column vector, for instance $X^1$, i.e. the first column of matrix $A_{11}X_{11} + I$ (corresponding to the first column on the left of the above equations), we get:

$$X^1 = \begin{bmatrix} x_{11} \\ x_{21} \\ \vdots \\ x_{k1} \end{bmatrix} = A_{11}X_1 + I^1, \text{ where } I^1 \text{ is the first column of the identity matrix I: } I^1 = \begin{bmatrix} 1 \\ 0 \\ \vdots \\ 0 \end{bmatrix}$$

Such a vector gives us the quantities of the various commodities used as inputs in order to produce the quantity $x_1$ of commodity 1 which needs to be produced by the system to satisfy the final demand of one unit of commodity 1.

As for vector $X_I$, as defined above, giving the total quantities of all of the intermediate commodities to be produced by the system, in order to produce one unit of each intermediate commodity as “final demand”:

$$X_1 = \begin{bmatrix} x_1 \\ \vdots \\ x_{k1} \end{bmatrix}, \text{ we can determine it easily by using the summation vector } I_1, \text{ such that:}$$

$$X_1 = X_{11}I_1 + I_1$$
Once the outputs $x_{1i}, x_{2i}, \dots, x_{ki}$ (the $i^{th}$ column of matrix $X_{11}$) used to produce one unit of intermediate commodity $i$ are determined, it is easy to calculate the corresponding values, using their definition through this second method, given by equation (10) above:

$$v_i = \sum_{j=1}^{k} l_j x_{ji}$$

From this equation we get the values, i.e. in matrix notation the row vector $V_1$:

$$V_1 = L_1 X_{11} \qquad (22)$$

However it is clear that the $k$ equations from which we started our demonstration above do not include all the equations in which the first $k$ intermediate commodities of the first type intervene, and therefore do not include all the uses of these first $k$ commodities, which also enter into the production of the other $n-k$ final commodities.

Turning then to the production of these final commodities, we must add, for each intermediate commodity $i$ entering in the production of a final commodity $j$ (with $j = k+1, \dots, n$):

- Firstly, a share $a_{1j}, \dots, a_{kj}$ of the production of each intermediate commodity $l$ to $k$, that will be directly required to produce one unit of each final commodity $j$ (with $j = k+1$ to $n$), for instance $a_{i,k+1}$ for the quantity of intermediate commodity $i$ entering directly in the production of final commodity $k+1$.

- Secondly, the indirect quantities of this intermediate commodity embodied into all of the other intermediate commodities entering in the production of this final commodity;

Therefore for each final commodity $j$ we can write a series of equations, which for a final commodity $j$, is the following:

$$x_{1j} = a_{11}x_{1j} + \dots + a_{1k}x_{kj} + a_{1j}$$

$$x_{2j} = a_{21}x_{1j} + \dots + a_{2k}x_{kj} + a_{2j}$$

$$x_{kj} = a_{k1}x_{1j} + \dots + a_{kk}x_{kj} + a_{kj}$$

Similarly, for $j = k+1$ and for $j = n$, these sets of equations are:

$$x_{1,k+1} = a_{11}x_{1,k+1} + \dots + a_{1k}x_{k,k+1} + a_{1,k+1}$$

$$x_{2,k+1} = a_{21}x_{1,k+1} + \dots + a_{2k}x_{k,k+1} + a_{2,k+1}$$

$$x_{k,k+1} = a_{k1}x_{1,k+1} + \dots + a_{kk}x_{k,k+1} + a_{k,k+1}$$

and

$$x_{1n} = a_{11}x_{1n} + \dots + a_{1k}x_{kn} + a_{1n}$$

$$x_{2n} = a_{21}x_{1n} + \dots + a_{2k}x_{kn} + a_{2n}$$

$$x_{kn} = a_{k1}x_{1n} + \dots + a_{kk}x_{kn} + a_{kn}$$
With $\sum_{j=1}^{n} a_{ij} = 1, \forall i$, meaning that there is no surplus of any intermediate commodities $l$ to $k$.

In matrix notation, these additional equations for all $n-k$ commodities correspond to the multiplication of matrix $A_{11}$ by matrix $X_{12}$, this last one being a rectangular matrix of dimension $k \times (n-k)$, to which is added a matrix $A_{12}$, obviously of the same dimension.

Going back to the values of the $n-k$ final commodities, and to the second definition of these values, which says that they are the sum of the direct labor time used to produce the $k$ intermediate commodities entering into their production, plus the direct labor time used in their own production, i.e. in the transformation of these same intermediate commodities used in this process into final commodities, we get therefore:

$$v_j = \sum_{i=1}^{k} l_j x_{ij} + l_j \qquad (23)$$

Thus in matrix notation, we get the following equation for matrix $X_{12}$:

$$X_{12} = A_{11}X_{12} + A_{12} \qquad (24)$$

This gives us the solution for matrix $X_{12}$:

$$X_{12} = (I - A_{11})^{-1} A_{12} \qquad (25)$$

Now that both matrices $X_{11}$ and $X_{12}$ have been determined, we can finally calculate also the values, for the $n-k$ final commodities:

$$V_2 = L_1 X_{12} + L_2 \qquad (26)$$

## 4. The equivalence of the two methods of value determination

To begin with, it is quite easy to show that the two methods that have been considered successively arrive at identical results for the calculation of values. To demonstrate it, it is sufficient to replace $X_{11}$ in equation (22) giving the values of intermediate commodities with the second method, by its expression in equation (21). Then we get:

$$V_1 = L_1 (I - A_{11})^{-1} \qquad (27)$$

We note that equation (27) is identical to equation (8) which gave the values of intermediate commodities of the first type with the first method.
Similarly, if we replace in equation (26) giving the values of final commodities with the second method $X_{12}$ by its expression in equation (25), we obtain:

$$V_2 = L_1(I - A_{11})^{-1}A_{12} + L_2 \quad (28)$$

And we note again that equation (28) corresponds to what would happen of equation (7), which gave the values of final commodities with the first method, if there were no intermediate commodities of the second type, meaning that vector $V_2$ would not exist as a vector of values of intermediate commodities of the second type, but would be replaced by a new vector $V_2$ representing the values of final commodities.

## 5. The equivalence of total direct labor time and total product value

These same equations allow us to show also that the value of the product is equivalent to total direct labor time. To demonstrate it, let us call $y_1$ and $y_2$ the column vectors of the output of intermediate goods and final goods, respectively.

Then, from equations (8) and (9) we can write that the value of the total product is equal to:

$$V_1y_1 + V_2y_2 = L_1(I - A_{11})^{-1}y_1 + L_1(I - A_{11})^{-1}A_{12}y_2 + L_2y_2 \quad (29)$$

This expression, from equations (21) and (26), is equivalent to:

$$V_1y_1 + V_2y_2 = L_1X_{11}y_1 + L_1X_{12}y_2 + L_2y_2 \quad (30)$$

And this gives us indeed the overall quantity of direct labor time used in the production process, first for the production of intermediate commodities, i.e. $(L_1X_{11}y_1 + L_1X_{12}y_2)$, and second for the production of final commodities, i.e. $L_2y_2$.

It is also possible to show that the value of the product is equivalent to the value of intermediate commodities used as inputs in the production process of both intermediate and final commodities, plus the quantity of direct labor time utilized in the production of the “surplus” of intermediate commodities and of final commodities.

Taking again into account the fact that we assume here that there are no intermediate commodities of the second type, equations (5) and (6) give us indeed directly:

$$V_1y_1 + V_2y_2 = (V_1A_{11} + L_1)y_1 + (V_1A_{12} + L_2)y_2 \quad (31)$$

$$\Rightarrow V_1y_1 + V_2y_2 = (V_1A_{11}y_1 + V_1A_{12}y_2) + (L_1y_1 + L_2y_2) \quad (32)$$

Bringing equations (30) and (32) together also allows us to see that the value of intermediate commodities, used as inputs, which is $V_1A_{11}y_1 + V_1A_{12}y_2$, is equal to the difference between
the totality of direct labor time utilized in the production of all intermediate and final
commodities, which is $L_1X_{11}y_1 + L_1X_{12}y_2 + L_2y_2$, i.e. total employment, on the one hand,
minus the direct labor time utilized in the production of $y_1$ and $y_2$, i.e. $L_1y_1 + L_2y_2$, on the
other hand.

From these findings we can derive a few teachings about values, which will be exposed in
next chapter.

References

Miller, Ronald E. and Blair, Peter D (2009). *Input-Output Analysis: Foundations and Extensions*. Cambridge (UK): Cambridge University Press.

Morishima, Mishio (1973). *Marx's Economics: a Dual Theory of Value and Growth*. London: Cambridge University Press.

Peterson, Bruce and Olinick Michael (1982). *Leontief models, Markov chains, sub-stochastic matrices, and positive solutions to matrix equations*. *Mathematical modelling, An International Journal*, Vol. 3, pp. 221-239.
# Chapter 7. A few teachings about the concept of value

At this point there are some preliminary conclusions that we can draw in this chapter on the relevance of the concept of value and the conditions for this relevance, the first one having to do with the field of validity of this concept. We will also come back to the determination of values, to give a little more precisions on the conditions which make it possible. We will then be in a position to examine more closely the distinction between absolute values and relative values, which will help us to define the invariable standard of value. This will bring us to the mistakes made by Marx in the exposition of his labor theory of value, since they may partially explain some of the various criticisms still surrounding this theory. These criticisms, often based on misconceptions, will finally be discussed.

## 1. The field of validity of values

The first condition for the relevance of the concept of value has to do with its field of validity. What has indeed been shown so far is that this concept can be defined at the sole level of production, which means that values of commodities preexist, logically and chronologically, to the exchange of commodities on markets. Exchange of commodities as such plays therefore no role in the determination of values, which cannot be considered as exchange ratios, even though it is clear that these values will play a role in the determination of prices, defined as the exchange ratios of commodities against money, as we will see later.

Does it mean also that values can be determined as soon as there is production, whatever the underlying mode of production (which should be rather designated as a mode of distribution)? We should not go as far as that, since it has also be shown that the concept of value is very closely linked to and depends on another concept, that of abstract labor, which allows to consider labor purely in its dimension of time. Therefore it is only when abstract labor exists as such that the concept of value can remain a relevant one. This has to be related to one particular type of exchange whose conditions are determined before production takes place, and which is the exchange of labor against money, involving a definite quantity of labor time against a definite quantity of money.

To be sure, such an exchange is the case in the capitalist mode of production, where it is generalized, with most of the production made by wage-earners whose wages are indeed paid in money and defined as a number of monetary units for a specified quantity of time. The conditions of this exchange are determined before labor is executed, when workers are recruited. In turn this implies that values have a sense each time that there is paid labor and that this payment is determined and established as a specified amount of money for a specified quantity of time. To the extent that this could also take place, although to a more limited scale, in other modes of production, it would then certainly be relevant to also use the concept of value, when analyzing production relationships in these other modes of production.
## 2. The determination of values is possible and feasible

As we have seen, and on the bases of assumptions made in section 1 of chapter 6 (see p. 117), and which can be considered as empirically plausible, it has been possible to calculate values through two slightly different methods which were shown to be mathematically equivalent. These values are all positive (the reverse would be devoid of any significance) since the mathematical theory of matrices demonstrates that with any non-negative productive matrix such as matrix $A_{11}$ as defined (which means economically that each of the first k branches provides a surplus of intermediate commodities), we will always get positive solutions for the inverse matrix $(I - A_{11})^{-1}$. On this point the reader can refer again to the book by Peterson and Olinick, which was cited previously.

A second condition for values to be positive, since they are given by $V_1 = L_1(I - A_{11})^{-1}$, and $V_2 = L_1(I - A_{11})^{-1}A_{12} + L_2$, is obviously that labor has to be indispensable to the production of each of the intermediate commodities, which implies that the labor vector $L_1$ be strictly positive ($L_1 > 0$).

These two conditions allow to deduct that means of production which are not produced, i.e. essentially land in its original state, have a zero value, which does not mean that they have a zero price, but loosens somewhat assumption 3 made previously (see p. 117), because primary factors other than labor time cannot appear in the equations.

Mathematically speaking, this assumption of a strictly positive vector $L_i$ can however be released to the extent that we would only have $L_1 \ge 0$, on the condition that matrix $A_{11}$ be not only productive, but also indecomposable, which means economically that each of the intermediate commodities must enter into the production of at least one intermediate commodity other than itself, or reciprocally that every intermediate commodity must be produced by at least one intermediate commodity other than itself. We know that it is the case for all intermediate commodities of the first type, by their very definition. In this case the value of an intermediate commodity will be positive, even though no direct labor is spent in its production. The indecomposable nature of matrix $A_{11}$ and the fact that $L_1 \ge 0$ (one of the $l_i$ is at least positive), ensure indeed that the production of each commodity requires at least some indirect labor.

As for final commodities, the positivity of their values requires that matrix $\begin{bmatrix} A_{12} \\ L_2 \end{bmatrix}$ formed by the junction of matrix $A_{12}$ and vector $L_2$ be non-negative and that none of its columns be empty. This means that any final commodity must be produced with either labor or an intermediate commodity.

From all this it derives that the calculation of values for both intermediate and final commodities is perfectly feasible, on the basis of data which are usually available, of
assumptions which are empirically rather realistic ones and of conditions which are usually
met. This theory is therefore economically significant and relevant, and will appear even more
so if we address now the question of the distinction between absolute and relative values.

3. The distinction between absolute values and relative values and the invariable standard of value

It seems useful to devote a few lines to the distinction between absolute values and relative values, because some authors consider indeed that the system of values allows only for the calculation of *n* “absolute values”, understood like quantities of labor time, whereas the system of prices would allow for the calculation of only *n* − 1 prices. This would create some difficulty in the transformation of values into prices.

Let us recall first that as far as the system of values is a measurement system with the dimension of time, the values that we obtain are always relative to a standard of measurement which cannot be anything else than an arbitrary unit of time, such as one hour, one day, one week, one month, or one year, etc.). Then it is true that we can have *n* values for *n* commodities, each one being a certain amount of labor time measured in terms of this chosen unit of time. But it is also obvious that if the quantity of time corresponding to the value of one commodity is chosen as the measurement unit (or standard of value) - and we are free to do so, then there will be only *n* - 1 values different from 1 which could still be considered as “absolute values”, i.e. as specified amounts of labor time. Conversely, the price system can provide us with *n* prices different from 1, as soon as a basket of commodities (similar for instance to Sraffa’s standard commodity) is taken as the measurement unit.

The lesson that we can draw from these considerations is that in fact every system of
measurement always defines as many measurements as there are elements in the set to be
measured, and therefore the fact that one of these measurement is or is not equal to l is a pure
question of definition of this measurement unit, that can be internal or external to the set of
elements to be measured (provided however that it has the same dimension as these elements).
All this has no importance other than practical from an economical point of view.

Indeed this comes down to a question of vocabulary, being understood that usually when we talk about a certain magnitude, for instance a mass, we specify, if not the dimension of the measured element itself, at least implicitly the dimension of the unit of measurement: we will not usually say that such luggage has a mass of 15 kg, but that it weighs 15 kg, because we know that a kilogram is the measurement unit for the dimension of mass (and even the standard of measurement). We could say more accurately that its relative mass is 15, in terms of the measurement unit of mass, but it is not customary to say that.

These precisions being given, we must recall that there is nevertheless one fundamental
divergence between the measurement of exchange ratios and the measurement of values. As
we saw in the chapters on Ricardo and Sraffa the measurement of exchange ratios (i.e. for
these authors the prices of production) can never be compared when exchange ratios are
modified, because the measurement unit itself is defined in relation to a given state of these exchange ratios, depending itself on the state of techniques and on the state of distribution.

On the contrary the measurement of values can always be compared when they are modified, because they are defined either in terms of an invariable measurement unit (if it is a given quantity of labor time, chosen as the measurement unit), or in terms of measurement units that can themselves be compared (if the value of whatever commodity is chosen as the measurement unit, the modifications of this value can themselves be measured in terms of a given quantity of labor time). The measurement of values therefore refers to an absolute measurement unit, or a measurement unit which can always be expressed in terms of an absolute standard, for instance one hour of labor time. It is only in this well-defined sense of "measurement in terms of an absolute standard" that we can speak of "absolute values", if we think that it is necessary, although it is usually not the case. Let us recall that Marx himself does not use the expression "absolute value".

As for the expression "relative value", it is used by Marx in the restricted case where the measurement unit of values is not a quantity of labor time chosen arbitrarily and invariable, but the variable value of a given commodity, chosen as a standard commodity. Although values are as much relative when they are determined in terms of a given and fixed quantity of labor time, the expression "relative value" can indeed be considered as appropriate in this last case: indeed values are then twice relative, since the measurement unit itself can change, even though these variations may themselves be measured.

One has to be quite rigorous in trying to define laws of variation for this "relative value", and Marx himself makes some mistakes when he does so. In section 3 of chapter 1 of "Capital" (volume 1), he considers indeed (in point b) "quantitative determination of relative values", pp. 36-37):

1) That the "relative value" of a commodity rises and falls directly according to its own value (in terms of an invariable quantity of labor time), the value of the standard commodity B being constant.

2) That if the value of a commodity A is supposed to remain constant, its "relative value" expressed in the standard commodity B rises and falls inversely as the value of the standard commodity B.

3) That if the values of all commodities rise and fall simultaneously, and in the same proportion, their "relative values" would remain unaltered.

4) That when the values of a commodity A and of the standard commodity B vary simultaneously in the same direction but at unequal rates, or in opposite directions, the effect of all these possible different variations on the "relative value" of commodity A are diverse and may be deduced from a combination of rules 1) and 2).

In fact Marx seems to forget his own definition of relative value, which implies that the value of standard commodity B cannot be considered as a constant, whatever may be commodity A
for which we contemplate a variation of its relative value. It is indeed enough for this
commodity A to be an intermediate commodity to entail that a variation in its value (in terms
of a quantity of labor time) results in a variation in the value of the standard commodity B,
and this whatever this last one is an intermediate commodity or a final commodity (even
though this final commodity is produced without using directly the first commodity). Then
neither rules 1) and 2) hold, no more than rule 4) which derives from them.

In fact, if we deal with this problem while taking into account this phenomenon, and in the
light of what we learned from the determination of values, the conclusions that we can reach
are more precise.

1) If we first consider the case where only the quantity of direct labor time used to produce a commodity *i* varies, while the direct labor time employed to produce all the other commodities remains constant as well as all the technical coefficients, we can only conclude that the relative value of commodity A in terms of the standard commodity is modified proportionally more than the relative value of any other commodity *j*. If commodity *i* is an intermediate commodity, the variation in its value will cause a variation in the value of all the other commodities, final commodities and intermediate commodities alike. If it is a final commodity, there will be no variation in the value of intermediate commodities, neither in the value of final commodities: it is the only case where rule 1) applies.

2) If however, the variation in the value of a commodity *i* comes from a technical improvement which reduces the quantity of one of its intermediate commodities, one can demonstrate equally that the value of commodity *i* diminishes proportionally more than that of any other commodity. Such a demonstration can be found in chapter 3 of Morishima's book (op. cit.). The same findings can be made concerning the distinction between the case where *i* is an intermediate commodity and the case where it is a final commodity.

All this discussion may seem somewhat esoteric, but in fact it is not, because it has allowed us
to discover that in the field of values we have found an invariable standard, which can be any
specified quantity of labor time, in terms of a specified time unit. This is something that had
been understood by Ricardo. And this invariable standard cannot be the value of a given
commodity, for the reasons which were just explained. This is of fundamental importance for
any economical reasoning, since it shows us at the same time what is a fundamental flaw for
both the neo-classical theory and Sraffa's theory of production prices: i.e. the contingent
nature of their measurement unit.

It must indeed be understood that in any theory all of the measured elements have necessarily
the same dimension as the measurement unit, which plays the role of the standard of prices in
both of these last two theories. In the neo-classical theory, all commodities have thus the same
dimension as the commodity that is chosen as money, and whose conditions of production
change continuously with the change of techniques and the changes in distribution. In Sraffa's
theory all commodities have the same dimension as the standard commodity, and even wages
and profits, as a share of the standard net product, are measured in terms of this standard commodity. The problem is that this standard commodity changes also continuously overtime, and is invariable only in a very narrow sense, to a change in the distribution which does not modify the techniques and structure of production.

On the contrary, we can consider that the labor theory of value, such as it has been exposed in the two previous chapters, has a truly invariable standard of value, which is an arbitrary quantity of physical time, and is a perfectly valid and relevant theory. As such this theory should be rehabilitated, despite the fact that it has been and is still so harshly criticized that it has fallen into desuetude. This rehabilitation would be all the more appropriate that these criticisms can be easily dismissed, as we shall see below. But we have to realize first that several mistakes made by Marx himself certainly facilitated the task of critics.

**4. The labor theory of value and Marx's mistakes**

In the light of everything that precedes, it is easy to show that Marx himself committed in the exposition of his labor theory of value a series of mistakes. To be sure, coming after the great classical economists Smith and Ricardo, he was in good company. But these mistakes, because of the confusion that they brought about in this complicated field, certainly helped in the rejection of the labor theory of value. This is why they will now be briefly recalled.

**4.1. A first and main error: an inaccurate vision of exchange**

The first and certainly the most fundamental error has to do with the very concept of value. Marx starts from exchange as an equivalence relation, but he conceives exchange as an equivalence relation directly between commodities. Then he considers as the only common property that allows to justify such an equivalence that commodities are the product of labor, and of abstract labor, and although he understands rightly that this equivalence between commodities can and must be established at the very level of production, he thinks that ipso facto this equivalence can be extended to the level of exchange, because he conceives it as an exchange between two commodities, one of them playing the role of money. Therefore he deducts in book one of Capital that exchange is an equivalence relation between values whose "substance" is labor whereas it is in fact labor time, i.e. time, which as such has no material content and is devoid of any substance.

He makes these mistakes despite the fact that he has correctly defined abstract labor as labor time, and because he does not from the start conceives exchange as a monetary exchange, i.e. as an exchange, not between two commodities, but between a commodity and money, this last entity having nothing to do with whatever commodity. This leads us to his second error.

**4.2. A second error: money as a commodity**

This can be considered in fact simply as a corollary of the previous error, since it derives in turn from the fact that when Marx introduces money, he immediately conceives money itself as a commodity, which entails that even monetary exchange is thought of as an exchange between two commodities, in the present case between any commodity and the commodity
that plays the role of money. Then it becomes impossible for Marx to establish a clear
conceptual distinction between an equivalence relation at the level of values and production,
on the one hand, and an equivalence relation at the level of exchange and prices, on the other
hand. For Marx this second equivalence relation at the level of exchange differs from the
previous one only because of some specific properties of the money-commodity. This leads us
to his third error.

**4.3. A third error: the common dimension of values and prices**

Because of the two last errors, values and prices belong for Marx to the same field of
measurement, which implies that prices, even though they are transformed values, still remain
comparable to values. What makes prices differ from values in Marx’s theory is only that it is
necessary to equalize profit rates on commodities whose conditions of production differ. But
values and prices remain values and prices which are relative to the value or the price of a
commodity chosen as a standard of value or prices (the standard commodity as money), with
the common dimension of corresponding to a quantity of abstract labor in Marx’s sense. It is
what explains that for Marx prices are production prices, and not monetary prices.

What has been shown so far, is precisely - and unlike Marx, that value, as an equivalence relation between commodities, cannot be an equivalence relation at the level of exchange, because precisely commodities do not exchange according to their value, and because there is no exchange between commodities, but between commodities and money. Value can therefore correspond to an equivalence relation only at the level of production, as long as production is made with wage labor. As for monetary prices, they have the same nature as money, which means that they are pure scalars, which do not have therefore the same dimension as values, and do to not belong to the same space (in the mathematical sense of this last term). Consequently it would be senseless to try and directly compare values and prices.

**4.4. A fourth error: the transmission of value by fixed capital**

The last error concerns the assumption made by Marx that fixed capital transfers its value to the product. This point has already been developed previously, therefore we will not expand too much on it. Suffice it to say that Marx could not master the scientific concept of dimension, which was developed first by Maxwell, who published in 1863 an article on the question, and supervised after 1866 the experimental determination of electrical units for the British Association for the Advancement of Science. His work in *measurement* and standardization led to the establishment of the National Physical Laboratory.

Therefore Marx could not grasp the fact that value, as a property of commodities, has the conceptual status of a dimension, which is not a detachable property of the commodities that possess it, and cannot be transmitted as a material substance might be (even though he sees abstract labor as a kind of substance of value - which is wrong too...). Similarly, it would be senseless to say that an object with a given mass could transmit and add its mass to another one virtually, without undergoing the same process of being weighted together.
The fact is that fixed capital, and in this regard it differs from intermediate commodities, is not transformed into commodities during the production process. Whereas intermediate commodities disappear in their original form in the course of this process, fixed capital, and for instance the machines, remain unchanged during it, because they are essentially mere operators or catalysts of the transformation that intermediate commodities undergo. As such fixed capital is obviously indispensable to this process, where it plays a fundamental role, because production could not take place without it. But in itself it does not create anything more, and in particular it does not create any more surplus than a catalyst usually does: it facilitates and quickens the transformation of intermediate commodities into different and more elaborate commodities, be they other intermediate commodities or final ones.

# 5. Some misconceptions about the labor theory of value

The labor theory of value has been much criticized over one and a half century. However, as we shall see, most of these criticisms are innocuous, because they are based on some kind of misunderstandings: this justifies that we consider them as elementary misconceptions, which will be addressed first in the next sub-section. This being done, a more serious criticism, developed initially by Arun Bose and based on the theory of values as dated quantities of labor, will be examined in a second sub-section, where it will be shown that it can also be dismissed on theoretical grounds.

## 5.1. Elementary misconceptions

The usual misconceptions are based on examples of individual commodities or special circumstances showing that monetary prices are greatly divergent from labor values: it can concern for instance a diamond found by chance without any real work, differences in productivity of workers that have no influence on values, the high price of works of art for sometimes very little work, or on the contrary the large amount of labor time that can be spent on particular occasions to produce some particular kind of goods that are useless. Some comparisons concern old goods that had a high labor value in the past but have no more now.

These criticisms do not take into account the fact that commodities are produced in large numbers, by abstract labor, i.e. labor which is not only reduced to its only dimension of time, but is also socially necessary. Consequently values correspond to an average labor time performed with average skills and an average productivity. This point was emphasized by Marx, who wrote: "Some people might think that if the value of a commodity is determined by the quantity of labor spent on it, the more idle and un-skillful the laborer, the more valuable would his commodity be, because more time would be required in its production. The labor, however, that forms the substance of value, is homogeneous human labor, expenditure of one uniform labor power. The total labor power of society, which is embodied in the sum total of the values of all commodities produced by that society, counts here as one homogeneous mass of human labor power, composed though it be of innumerable individual units...The labor time socially necessary is that required to produce an article under the normal conditions of production, and with the average degree of skill and intensity prevalent at the time" (Marx, K., 1867, Chapter 1. Section 1, p. 29).
Another quotation of Marx on the question, much further in “Capital” deserves to be noted, because it brings about even more precision to the previous statement, and shows well that the notion of an individual value is irrelevant in his theory. This quotation is the following: “the real value of a commodity is, however, not its individual value, but its social value; that is to say, the real value is not measured by the labor-time that the article in each individual case costs the producer, but by the labor-time socially required for its production. If therefore, the capitalist who applies the new method, sells his commodity at its social value of one shilling, he sells it for three pence above its individual value, and thus realizes an extra surplus value of three pence” (Marx, K., 1867, Chapter 12, p. 223).

It follows that values (like in fact prices) are average values (or average prices), established over large quantities of similar or fungible products, which have been produced by a number of producers, possibly with different techniques or methods of production. These products must themselves be useful, or in Marxian vocabulary, must have a use value. Values are also meaningful over time, since comparisons at different periods help to understand that the use-value of some commodities has decreased due to innovations that occurred later, and have made them obsolete, or that technical progress and increased labor productivity has reduced their value.

**5.2. Arun Bose's criticism: values as dated quantities of labor and the alleged existence of a commodity residue**

**5.2.1. A criticism based on Sraffa's theory**

Arun Bose is an Indian economist whose criticism of Marx's theory of value was formulated in a book published in 1980, and entitled “Marx on Exploitation and Inequality”. His criticism must be taken seriously, if only because it has been endorsed by Steve Keen, who in his previously cited book, indicates: “Bose argued...that if value was the essence of a commodity, then that essence could not be reduced solely to labor. Therefore, labor alone was not the essence of value: instead, both labor and commodities were the essence of value. His logic used a concept we saw earlier in chapter 6 (of Keen's book A.N.) the reduction of commodity inputs to dated labor” (Keen, 2011, pp. 431-432).

In passing let us recall that value is not, as we have shown, the “essence” of a commodity, but one of its many dimensions, in the sense that the word “dimension” has in physics. The word “essence” indeed is a kind of philosophical notion but not a scientific concept and has no theoretical meaning. This being said, Bose's criticism is indeed strictly based on chapter VI of Sraffa's book, a chapter which is entitled “Reduction to dated quantities of labor”. This explains that we prefer to start our demonstration by first going back directly to “Production of Commodities...”, and more precisely to this particular chapter, which we did not discuss specifically so far.

**5.2.2. Reduction to dated quantities of labor in Sraffa's theory**

In chapter VI of his book Sraffa defines “reduction to dated quantities of labor” as “an operation by which in the equation of a commodity the different means of production used are
replaced with a series of quantities of labor, each with its appropriate ‘date’” (Sraffa, 1960, § 46, p. 40). Sraffa’s demonstration starts with the equation giving the production price of commodity “a”, with the wage and prices expressed in terms of the Standard commodity:

$$ (A_a p_a + B_a p_b + \dots + K_a p_k)(1+r) + L_a w = A p_a \quad (1) $$

Sraffa then replaces “the commodities forming the means of production of A with their own means of production and quantities of labor, i.e. with the commodities and labor which, as appears from their own respective equations, must be employed to produce those means of production; and they, having been expended one year earlier (§ 9), will be multiplied by a profit factor at a compound rate for the appropriate period, namely the means of production by $(1+r)^2$ and the labor by $(1+r)$” (Sraffa, 1960, p. 40).

It must here be recalled that in § 9 Sraffa “assumes that the wage is paid post factum as a share of the annual product, thus abandoning the classical economists’ idea of a wage advanced from capital. We retain however the supposition of an annual cycle of production with an annual market.” (Sraffa, 1960, p. 11).

Sraffa repeats the same operation, and next to the direct labor $L_a$ he places “the successive aggregate quantities of labor which we collect at each step and which we shall call respectively $L_{a1}, L_{a2}, \dots, L_{an}, \dots$” (Sraffa, 1960, p. 41). By summation for each period he thus obtains what he calls ‘the reduction equation’ for the product in the form of an infinite series:

$$ L_a w + L_{a1} w(1+r) + \dots + L_{an} w(1+r)^n + \dots = A p_a \quad (2) $$

Sraffa then continues as follows: “How far the reduction needs to be pushed in order to obtain a given degree of approximation depends on the level of the rate of profit: the nearer the latter is to its maximum, the further must the reduction be carried. Besides the labor terms there will always be a ‘commodity residue’<fnref n="6" /> consisting in minute fractions of every basic product; but it is always possible, by carrying the reduction sufficiently far, to render the residue so small as to have, at any prefixed rate of profits short of R, a negligible effect on prices. It is only at $r=R$ that the residue becomes all important as the sole determinant of the price of the product” (Sraffa, 1960, § 46, p. 41).

Let us state first that we are not at all here in the realm of production and values, because there are not only wages but also profits, which cannot be realized by capitalists before they have not only paid wages to the workers but also sold their products on the market. We are therefore in the realm of distribution and prices, and more precisely of production prices fixed in such manner that they allow these capitalists to earn an average rate of profits. It follows logically that a statement which could be true for prices might not be true for values. Let us recall also that we have already seriously criticized Sraffa’s theory. However and as

<footnote n="6">⁶ The italic characters are from the author.</footnote>
mentioned above, in doing so we had not touched directly upon this question of reduction to
dated quantities of labor, which explains why it still deserves to be discussed.

A first remark concerns Sraffa's position that wages are paid *post factum*, as he says, i.e. once some labor has been accomplished. We have previously explained the theoretical reason why he is obliged to do so, and we cannot but endorse this position, all the more so that it corresponds to what is the case in the real world, to which any theory should stick as much as possible in its assumptions. Indeed workers are usually and almost systematically paid after labor has been carried out, be it after one day, one week, a fortnight or most often one month. Wage is therefore not advanced, in the sense that it is not paid before labor has been executed.

This creates a logical problem in Sraffa's system, where production takes place during one full year without being sold, i.e. without any revenues for the producer, because there is “an annual cycle of production and an annual market” (see again § 9). Indeed workers cannot stay one year without being paid, and simply wait until the market day arrives! In passing this point shows the influence of the Physiocrats on Sraffa's conception of production, which he acknowledges at the very end of his book, in appendix D “References to the literature” (Sraffa, 1960, p. 111): it is as if we were in a world of agricultural production only, with one harvest per year, and an annual market afterward.

All this means that if wages are paid in due time and are therefore not advanced from the point of view of the workers, they have to be advanced nevertheless from the point of view of the capitalists. Then, if workers start being paid from almost the beginning of the year (say after one week or one month of labor), why should wages be treated differently from the means of production bought at the beginning of the year and why should they not earn the producer the same profit rate, *prorata temporis*, as these means of production? Or conversely, if wages paid during almost the totality of the first year preceding the market do not earn this profit rate, why should the means of production employed simultaneously during this same period earn it? This is quite a logical problem, left unsolved by Sraffa, who does not even mentions it.

In fact these logical difficulties all come from Sraffa's erroneous conception of production, which he sees as a discontinuous and discrete process: even if he refers no more to agricultural production but to industrial production, it is as if all producers had to buy at the beginning of each year the full stock of means of production that will be processed all along the full year to come! Obviously this assumption has nothing to do with the real world, in which production is carried out more and more on a just-in-time basis with minimal stocks to avoid storage costs and constraints. Moreover in any case the only fact that production takes place means that the necessary intermediate commodities are available at any point in time.

Anyhow, and because of his wrong conception of production as the creation of a surplus which materializes once a year, what Sraffa misses completely is that production is fundamentally a continuous process, with extremely short production periods. These periods have recently been even more shortened by the progress of robotization: an automobile plant currently produces more than 1000 cars a day, or a steelworks more than 10 000 tons per day.
At this stage we can better understand that Sraffa's theory of dated quantities of labor boils down to a revival of the Austrian theory of production, such as formulated by Eugen von Böhm-Bawerk, in "The Positive Theory of Capital", a book that he published in 1891. For this economist, due to the *roundabout* nature of production, which necessarily involves the passage of time, the return on capital arises from this detour of production and the risk that it involves. Sraffa does not seem to be aware that the Russian economist Nikolai Bukharin had stricken a powerful blow at Böhm-Bawerk's theory of production in his 1927 book: "Economic Theory of the Leisure Class". Bukharin recalled rightly that production is a continuous process, which could not take place at any stage if the necessary intermediate commodities were not fully available to be transformed in this process.

We could not say it better than Bukharin does: *"But in reality we need neither to "wait" nor postpone consumption, for the simple reason that the social product, whatever may be the section of production we are considering, is present simultaneously in all the stages of its manufacture, if we are dealing merely with a social production process. Marx already pointed out that the division of labor replaces the "succession in time" by a "succession in place."*
Karl Rodbertus describes the process as follows: "In all the 'enterprises' of all the branches of all stages of production, simultaneous uninterrupted labor is going on. While in the production establishments of the branches of raw production, new raw materials are being won from the earth, the production establishments in the branches of intermediate products are simultaneously transforming the raw materials of the preceding epoch into intermediate products; while the tool-producing factors are replacing tools that have been used up, and while, finally, at the last stage of production, new products are being turned out for immediate consumption (Karl Rodbertus: *Das Kapital*, p. 257. Berlin, 1884.)." (Bukharin, 1927, Chapter IV, section 3, § 5).

If we dig a little further into the arcana of Sraffa's theory, as we already did in chapter 4, it appears that all these theoretical difficulties that Sraffa encounters can be traced back to his wrong view of production as a circular process. This explains that we stand here in a system where we only have intermediate goods, which implies that wages are a share of a Standard net product made of intermediate commodities. Obviously this is absurd, because workers do not consume intermediate commodities, but consumption goods which are final commodities. Similarly it is absurd to imagine that the structure of this consumption does not change when wages change and when this share is modified.

A last reason why this notion of dated quantities of labor cannot be taken seriously is the following: however far we go back in the past when computing these quantities, we must assume that the Standard net product and the Standard commodity have not changed over the years, since otherwise the changes in distribution would be meaningless, being expressed in variable Standards. This implies that we must assume that the production system itself has remained the same over the years, keeping the same techniques and the same structure. This is absurd, but it is the theoretical ransom to be paid for having a Standard measurement unit which has is a basket of commodities, and not a truly invariable Standard. We could even be tempted to say that Sraffa's Standard measurement unit has the dimension of a basket of
commodities, although this would be faulty: a basket of commodities as such cannot have a dimension, in the scientific sense of the term! Moreover, when the rate of profits changes, equation (2) above shows that it has to be changed also for the preceding years, so as to remain exactly the same for the whole period, whereas in the real world the profit rate changes continuously overtime.

### 5.2.3. The absence of a commodity residue

We can now go back to the original question: is there or not, as Sraffa claims, a commodity residue? To answer precisely the question, let us remark first that in equation (1), which is Sraffa's point of departure, there are no commodities in isolation, since we stand in the space of production prices, and all the quantities of commodities are multiplied by their price in terms of the Standard commodity. Therefore a commodity residue must be understood as equivalent to a price residue. If we reformulate that in mathematical terms, it means that equation (1) will never provide us with an absolute price, because there would always remain - to paraphrase Sraffa – “a (price) residue consisting of minute fractions of (the price) of every basic product”.

To decide whether or not these production prices are approximate comes down to decide whether or not the equations giving the price of commodities converge. Hence we arrive to a manageable mathematical problem.

Indeed equations (1) and (2) in sub-section 5.2.1. correspond to Sraffa's system as described earlier in chapter 3, and can be synthetized in the following matrix equation (see above in section 3 of chapter 3), where P is the column vector of prices:

$$P = [I - (1+r)A]^{-1} wL \quad (3)$$

Where L is a column vector of labor quantities in each branch, as a fraction of total labor.

This equation is equivalent to:

$$P = [I + (1+r)A + (1+r)^2 A^2 + \dots + (1+r)^n A^n] wL, \text{ with } n \to \infty \quad (4)$$

The reason why this function converges must be demonstrated. Let us recall first what was seen in chapter 2: from the Perron-Frobenius theorem, which we used in this chapter (we recall that a demonstration is given by Gantmacher, previously cited, we know that the only eigenvalue of an irreducible matrix with non-negative terms to which corresponds a positive eigenvector is the dominant eigenvalue of this matrix, i.e. the highest one. Therefore we can write:

$AP = \alpha P$, where A is a non-negative, sub-stochastic and productive matrix and P is the price vector, which the Perron-Frobenius theorem guarantees is positive.
Secondly, from the construction of the invariable Standard commodity (see above in section 3
of chapter 3), we know that $\alpha = \frac{1}{1+R}$, with R being the maximum profit rate of this system.

It follows that $AP = \frac{1}{1+R}P$

Thirdly, since $w$ is expressed in terms of the Standard commodity, we know that $r$ must
necessarily be the independent variable, and that we have the relation: $r = R(1-w)$, which
means that for any value of $w > 0$ we also have $r < R$.

Therefore, for any such value of $r$, we have necessarily:

$AP < \frac{1}{1+r}P \Rightarrow (1+r)AP < P$, which implies that matrix $(1+r)A$ is also by definition a sub-stochastic and productive matrix. Moreover every positive power of a stochastic matrix is stochastic, a property which extends to sub-stochastic matrices, as is shown by Peterson and Olinick (op. cit.).

From this it follows that $(1+r)^n A^n \to 0$ when $n \to \infty$

It is now easy to show that there is no commodity residue in equation (4) above which gives
the price vector of the system and that the limit of the sequence of powers of matrices
corresponds to a price vector of which each element is itself a real number. In other words it
can be shown that none of these numbers is an approximate number, because the
approximation stays only in the notation. This assertion is supported by several
demonstrations. Since I am not a true mathematician, I prefer to borrow these demonstrations
from a recent article by the French mathematician Jean-Paul Delahaye, published in 2016 in
the French edition of *Scientific American*, and titled "Est-il vrai que 0,999... = 1 ?" ("Is it true
that 0.999...=1?"). This synthesis of well-known mathematical results definitively settles the
question, which explains that these short demonstrations are reproduced hereafter.

1) We know that $\frac{1}{3} = 0.333...$, because indeed it is what we obtain if we realize the division.

But if we multiply both members of the equation by 3, we obtain: $1 = 0.999...$, an equality
which shows that there is no residue between 1 and 0.999...!

2) Let us have $u = 0.999...$

Then let us multiply both sides of this equality by 10. We get: $10u = 9.999...$

Now let us subtract $u = 0.999...$ on both sides of this last equality. Then, because obviously
$9.999... - 0.999... = 9$, we get: $10u - u = 9 \Rightarrow u = 1$

We are thus back to the equation: $1 = 0.999...$, which shows again that there is no residue!
In fact this is so because numbers are nowadays conceived as limits of numerical converging sequences, and the notation for expressing that conveys the false impression that there would be a residue. All numbers are indeed the limit of a series where each number has a finite decimal expansion: 0.333 is the limit of the series 3/10, 33/100, 333/1000, etc.

3) Let us assume that 0.999... < 1, as if there were a residue, and consider the average $m$ of both numbers 0.999... and 1: it is a number which is less than 1 and greater than 0.999..., because the average of two numbers is always between these two numbers. The decimal writing of $m$ starts therefore by 0.9. This average is also greater than 0.99 and less than 1: it is therefore a number of which the decimal writing starts by 0.99. By continuing like that, we establish that the decimal expansion of $m$ is necessarily 0.999...

This implies that this average $m$ is 0.999..., i.e. the smaller of the two numbers. But it is absurd, because the average of two different numbers cannot by definition be equal to any of them. The only explanation to this paradox is that the starting assumption is wrong. Since it is equally impossible to have 0.999... > 1, we necessarily have $1 = 0.999...$ which means once more that there is no residue!

We could also have ended the demonstration by stating that $u = 0.999...$, and that by definition

$$m = \frac{(u+1)}{2} = u, \text{ which implies that } u+1 = 2u, \text{ which gives } u = 1, \text{ or } 0.999... = 1, \text{ as always!}$$

4) A last demonstration starts by writing that $u$ is the limit, when $n \to \infty$, of a series of which the general term is:

$$x_n = 0.9 + 0.09 + \dots + 0.0\dots09$$

We can then make the factorization:

$$x_n = 0.9 \left( 1 + \frac{1}{10} + \dots + \frac{1}{10^n} \right)$$

Then, knowing that $0.9 = 1 - \frac{1}{10}$,

We can use the well-known identity: $(1-a)(1+a+a^2+a^3+\dots+a^n) = 1-a^{n+1}$, where we replace $a$ by $\frac{1}{10}$ to obtain the value of $x_n$ such that: $x_n = 1 - \left(\frac{1}{10}\right)^{n+1}$

The series of term $10^{n+1}$ tends towards infinity when $n$ tends towards infinity, so that $\frac{1}{10^{n+1}}$ tends towards 0, and $x_n$ tends towards 1. Therefore we come again to the conclusion that $1 = 0.999...$, which shows once more that there is no residue!
To conclude on this question, all these demonstrations should have made it clear that for compelling mathematical reasons there is no such thing as a residue, when we are dealing with the convergence of numerical series, which is what this whole question comes down. It is hard to understand how Sraffa made such a mistake, and all the more so that, if it had not been a mistake, it is his whole theory of production prices that would have suffered from this approximate character of prices. Indeed prices as dated quantities of labor are only a modality of presentation of the fundamental equations of his system, and the criticism could have been extended to the whole system.

It is equally difficult to understand how Arun Bose, and unfortunately after him, Steve Keen, may have endorsed Sraffa's contention about a residue in § 46 of his book, without checking its validity. Maybe it could be so because both economists are Sraffians who adhere to and are convinced of the validity of the whole theory of production prices? Or more simply, not being true mathematicians, they are not familiar with the present mathematical definition of limits of numerical series, which is used with the theory of sets to construct the set of real numbers. Jean-Paul Delahaye also recalls that this conception of real numbers "allows to solve the problem of the comparison between 1 and 0.999...:

$$ 
\begin{align*}
0.999\ldots &= \lim_{n \to \infty} 0.99\ldots9 \quad (\text{with } n \text{'9'}) \Rightarrow 0.999\ldots = \lim_{n \to \infty} \left(1 - \frac{1}{10^n}\right) \\
\Rightarrow 0.999\ldots &= 1 - \lim_{n \to \infty} \frac{1}{10^n} = 1'' \quad (\text{QED}) \tag*{(Delahaye, 2016, p. 79)}
\end{align*}
$$

The number 1 is thus the limit of a series where each element has a finite decimal expansion, and so it is with production prices, that are real numbers, representing shares of the Standard commodity, which means that there is no such thing as whatever commodity residue!

### 5.2.4. Dated quantities of labor as a wrong concept

The important teaching that we can draw from this sub-section is not only that there is no such thing as a residue of commodities in the definition of prices through their reduction to quantities of labor, but also that there are no such things as dated quantities of labor, because production is a continuous process, and the whole production of a period is produced during the same period, including all of the intermediate commodities used during this period. Any possible inventories of commodities coming from the preceding period(s) must be added to the production of the period because anyway they will be transformed in final commodities of the period and at the very beginning of it, and similarly intermediate commodities unused and stored at the end of the period must be deducted from its product, which is made of final commodities only. The only thing which intervenes in the production process and can be considered as dated are the machines produced in earlier periods and still in use during the period, but anyhow they do not transfer any value to the product of any period.

Now that we have dealt with this question of residue of commodities in the realm of production prices, by demonstrating the absence of such a residue, it is going to be very easy
to address it in the realm of values. It is indeed immediate to reckon that if we start from equation (2) giving the production price of a commodity in sub-section **5.2.1**. in terms of “dated quantities of labor”:

$$L_a w + L_{a1} w(1+r) + \dots + L_{an} w(1+r)^n + \dots = A p_a$$

And if we remove the profits, and consequently the dates, then we obtain the following equation:

$$L_a w + L_{a1} w + \dots + L_{an} w + \dots = A p_a$$

If we now remove the wages w from this last equation, it will be exactly similar to the equation giving the value of a commodity $v_j$ with the second definition of values that we exposed in chapter 6 where there is only direct labor time, which includes the direct labor time used in this commodity's own production, i.e. in the transformation of intermediate commodities used in this final process, plus the direct labor time used to produce these same intermediate commodities, and in turn the direct labor time used to produce their own means of production, and so on:

$$v_j = x_{1j} l_1 + x_{2j} l_2 + \dots + x_{kj} l_k$$

With $L_a = x_{1j}l_1$, $L_{a1} = x_{2j}l_2$, $L_{an} = x_{n+1,j}l_{n+1}$, both equations are similar, except for the number of intermediate commodities, which can easily be fixed and does not matter.

If we now go back to the set of all the equations for all intermediate commodities, in matrix format, we get what was equation (12) in chapter 6, giving the value for the second method of calculation of these values:

$$V_1 = L_1 X_{11}$$

But we have shown that this was equivalent to equation (17), i.e.:

$$V_1 = L_1 (I - A_{11})^{-1}, \text{ which can be replaced by its expansion in a power series:}$$

$$V_1 = L_1 (I + A + A^2 + \dots + A^n) \text{ with } n \to \infty$$

Since it has been already demonstrated that this sequence of power matrices converges for prices when the general term of the corresponding series is $(1+r)A$, it is immediate that it is also convergent when the general term is A, i.e. a sub-stochastic and productive matrix. And we know that A has to be so, otherwise there would be no surplus of intermediate commodities left to be used in the production of final commodities.
It follows that there is no residue either in the calculation of values, which are real numbers in terms of the chosen measurement unit, which is a unit of physical time.

This criticism of the labor theory of value on the basis of an alleged commodity residue has thus been definitively put aside. It is therefore time now to address another and last criticism, which deserves a full chapter, since it is a serious one, and has to do with the problem of joint production.

## References

Böhm-Bawerk, Eugen (1891). *The Positive Theory of Capital* (Smart, W. A., trans.). London: Macmillan and Co.

Bose, Arun (1980). *Marx on Exploitation and Inequality: An Essay in Marxian Analytical Economics*. Delhi: Oxford University Press.

Bukharin, Nikolai (1927). Economic Theory of the Leisure Class. *Marxist Internet Archive* [online]. Marxist Economics Subject Archive. Nikolai Bukharin Archive. Available at: https://www.marxists.org/archive/bukharin/works/1927/leisure-economics/index.htm
Accessed 17 Jul 2018.

Delahaye, Jean-Paul (2016). *Logique & Calcul - Est-il vrai que 0,999 = 1?*. (Logic & calculation - Is it true that 0.999 = 1?). *Pour la Science* (French Edition of *Scientific American*), August 2016, N° 466.

Marx, Karl (1867). *Capital: A Critique of Political Economy*. Volume I - Book One: The Process of Production of Capital. Source: First English Edition of 1887. Marx/Engels Archive [online]. Available at: https://www.marxists.org/archive/marx/works/download/pdf/Capital-Volume-I.pdf
Accessed 25 Jul 2018.

Morishima, Mishio (1973). *Marx's Economics: a Dual Theory of Value and Growth*. Cambridge, London: Cambridge University Press.

Peterson, Bruce and Olinick Michael (1982). Leontief models, Markov chains, substochastic matrices, and positive solutions to matrix equations. *Mathematical modelling, An International Journal*, Vol. 3 (pp. 221-239).

Sraffa, Piero (1960) *Production of commodities by means of commodities. Prelude to a critique of economic theory*. Cambridge (UK): Cambridge University Press.
# Chapter 8. The labor theory of value and the problem of joint production

The criticism of the labor theory of value based on an alleged commodity residue has been definitively put aside in the previous chapter, and it is time now to address another and last criticism, based on the difficulties that the labor theory of value is supposed to encounter when it is confronted to the problem of joint production. In this chapter we will show that it is possible to solve this problem correctly.

The problem of joint production is a very tangled one, for a number of reasons which are going to be exposed in the present chapter. It is at the same time a serious one, judging by the fact that the alleged problems raised by joint production are the reason why Morishima, in the final chapter of his already cited book (Chapter 14: "The labor theory of value revisited"), put into question the validity of the labor theory of value under the form that he had exposed it so far, and proposed to revise it. He intended indeed to abandon the labor theory of value for a Von Neumann model of linear programming where equations are replaced by inequalities, and labor values are replaced by minimum labor requirements.

This problem of joint production is all the more complicated that there are several definitions of joint products, a point which we will examine first, noting that these definitions correspond partly to several levels of description of the real world and different theories of values and prices, which should not be unduly mixed.

## 1. Joint production and the various types of joint products

The whole difficulty of the matter comes from the fact that joint products cannot easily be distinguished from by-products. Things would be simple if everybody accepted a strict definition of a joint product as a product that results jointly with other products from processing a common input or several ones, with the proportions of these inputs going to each product impossible to distinguish. But in practice this is not the case, even if we turn to the *United Nations System of National Accounts (Studies in Methods)*, updated in 2014. This system of accounts offers the possibility to use a matrix-form of presentation of the accounts, which implies to resort to input/output tables. The methodology to develop such I/O tables is explained in an associated *Handbook of Input-Output Table - Compilation and Analysis*, last updated in 1999. This handbook distinguishes, among the "secondary products" resulting from production technology, three distinct types of products:

a) "Exclusive by-products, or products that are not produced separately anywhere, e.g. molasses linked to the production of sugar, new scrap in metal industry;

b) Ordinary by-products, or products that are technologically linked to the production of other products but are also produced separately elsewhere as main products. An example of this is hydrogen produced as a by-product in petroleum refining
establishments, but also produced separately by other establishments in the chemical industry;

c) Joint products, or products that are more loosely linked technologically than ordinary by-products. The common costs shared by joint products are more significant in value than is the case for ordinary by-products. One example is milk and meat in the livestock industry which may be produced in a scale that depends on the demand for each product and the ratio of the two products may be varied in response to changing conditions of demand. One of the joint products may be produced separately elsewhere. Joint-products cannot be easily distinguished from by-products" (United Nations, Handbook, 1999, p. 77).

However, when we leave the area of statistics to turn to a purely theoretical definition, when Sraffa addresses the question of joint production in Chapter VII of "Production of commodities..." his definition is more straightforward, since he writes: "We shall now suppose two of the commodities to be jointly produced by a single industry (or rather *by a single process*, as it will be more appropriate to call it in the present context)" (Sraffa, 1960, p. 51 - italic characters are from the author). The definition in this very last sentence is much more precise and in fact reduces the category of joint products to the above a) category, i.e. that of exclusive by-products. This means products which share not only one common input, but all of their inputs, and share them as already noted in proportions that are impossible to distinguish.

In the theoretical world this make sense, because if we suppose that in an otherwise common process of production for two different products there is only one input (either a commodity or a quantity of labor) for which the proportion going to each product can be distinguished, then the two corresponding production processes will be different, as little as this difference may be. Translated into mathematical language, it means that it will be possible to have two formally distinct and therefore independent equations: one for each process, and therefore that we will be no longer in the area where joint production creates a problem for the determination of individual values, i.e. when we have one equation corresponding to one single process for two unknowns values to be determined, but rather in an area where we can write two distinct equations.

In any case the linkage made by Sraffa between industry and process is quite appropriate, and also interesting because it draws our attention to the complexity of the real world.

## 2. The complexity of the real world as regards production

In the real world, or more precisely in the perception that we have of the real world, there are millions of producers. Some of them are small producers, among whom we can distinguish pure individuals who do not have a single employee and cannot be called capitalists in the usual sense of the word. There are also entrepreneurs who hire a few employees, small and medium enterprises, and bigger firms or companies, some of them, the largest ones, having more than a hundred thousand people on their payroll.
It is first in the agricultural sector, where there are sometimes millions of farmers in a single country - each of them producing but a few agricultural products, that there are quite certainly more producers than products, in spite of the many varieties which exist for each agricultural product. And although production processes for one particular agricultural product may not differ greatly, they certainly often differ from one producer to another, due to particular circumstances such as the differences in land quality.

Outside of agriculture anyway, although the number of producers for each particular product is usually smaller, most producers share the common characteristic of also producing several different products each, some of them being specific to only a few producers, and some being produced by many producers, although they may be sold under different brands.

Moreover, a number of producers may produce the same commodity, but usually each producer will have its own process of production, even though each process is only slightly different from the other producers', if only because the machines used by each producer are not exactly the same, or do not have the same age, or are not maintained similarly, which may affect the use of various inputs. A single producer may also use different processes, e.g. in different establishments, workshops or factories, for the same product. As a consequence, there are also millions of production processes, certainly more than the number of commodities themselves. The opposite case is in fact that of only one single producer for one product, which is the exact definition of a pure monopoly, and is not so frequent!

**3. The statisticians view of production and joint production**

All these various cases or circumstances exist in the real world and make a very complex picture, so that we could not build up a theory of production if we were to stay at this level of complexity. To reduce this complexity to a manageable theoretical level let us first turn to some degree of abstraction, and at the outset to statisticians. In order to describe the production process in an intelligible way, they use a number of simplifying techniques. First they compile data using as the statistical unit the establishment, which is a production unit consisting of either “an enterprise, or a part of an enterprise, that is situated in a single location and in which only a single, or (non-ancillary) productive activity ... accounts for most of the value added” (SNA, para. 5.21). However an establishment may engage “in one or more secondary activities, [which] should be on a small scale compared with the principal activity” (SNA, para. 5.22).

Once statisticians get the data corresponding to all of the establishments, they compile them in order to get the production accounts of “industries”. According to the SNA definition, an industry consists of a group of establishments engaged on the same, or similar, kinds of production activity. These activities are classified in categories, according to the UN International Standard Industrial Classification (or ISIC) which contains 17 major sections, 60 divisions, 169 groups and 291 industries (with four digits coding). More information on this question can be found in the *Handbook of Input-Output Table* (op. cit. p. 42).
Similarly, statisticians have to reduce the thousands of products of an actual productive
system to a meaningful and manageable number: this is achieved through a statistical unit for
products which is a unit of homogeneous goods and services, such as they appear in the
"Central Product Classification" (or CPC). This classification is quite exhaustive, with all
products being mutually exclusive. The detailed classification of products, which are either
outputs of domestic production activities or imports from non-resident sources, consists of 10
sections, 69 divisions, 291 groups, 1036 classes, 1787 sub-classes and can accommodate up to
65,610 categories. The principles for classification used by CPC are the following:

a) "For transportable goods, categories of products should be based on the physical properties and the intrinsic nature of products, i.e. the raw materials of which they are made, their stage of production, the use they are intended for, the prices at which they are sold, whether or not they can be stored, etc.",

b) "Individual goods and services as far as possible should contain only goods and services which are produced by a single industry" (United Nations, 1999, p. 42).

As we can see, there can be therefore in these statistical data an important discrepancy
between the number of industries and the much higher number of categories of commodities,
which is contrary to our initial finding that in the real world there should be more industries,
in the sense of production processes, than products. For statisticians, the matrix in which a
column shows for a given industry the amount of each commodity it uses as an input is
generally called the "use" matrix, and will have therefore more rows (the commodities) than
columns (the industries). It will be a rectangular, commodity-by-industry matrix of dimension
(n x m) and of rank n. Similarly the matrix in which a column shows for a given commodity
the amount of it produced by each industry, generally called the "make" matrix, will be a
rectangular, industry-by-commodity matrix of dimension (m x n) and of rank n.

Bearing in mind that statisticians as well as theoreticians have a common purpose of using the
production accounts or equations to perform some statistical or mathematical operations, then
their objective is to obtain symmetrical input-output tables, of either product-by-product or
industry-by-industry tables. Indeed such tables will allow statisticians to work with square
matrices, which are usually invertible, since only a square matrix can be inverted to obtain
what is usually called the Leontief inverse matrix. For statisticians it means that a
correspondence between ISIC and CPC systems of classification is needed: this is achieved,
but at a price, which is the existence of "secondary products", since their classification
contains more products than industries. Indeed in I-O accounting, the primary principle is that
each industry is associated with a commodity that is considered as the primary product of that
industry, and all other commodities produced by this same industry are considered as secondary
products.

For instance in the United States system of national accounts, which is certainly one of the
best systems currently available, since year 1997 I-O tables have incorporated a new
classification structure known as the North American Industry Classification System
(NAICS). NAICS was updated for 2002, and this corresponding NAICS has 20 major sectors
with a total of 1,179 industries. The US system and I-O tables are described by Horowitz and Planting in a book published in 2006 and updated in 2009 by the Bureau of Economic Analysis, US Department of Commerce: *Concepts and Methods of the US Input-Output Accounts*.

In spite of the high number of industries in these accounts, there are nevertheless a number of secondary products, because with the first principle above as the main criterion, different products may come out of the same industry. To give but one example provided by the CPC, published in the Statistical Papers of the United Nations: "For example, meat and hides are both produced by slaughterhouses. These products are not listed together in one category or even in the same section of the CPC. Unprocessed hides are considered raw animal materials, and they are classified in section 0 (agriculture, forestry and fishery products), whereas meat is classified in section 2, among food products" (United Nations, 2015, p. 8). It must thus be clear that most of these secondary products of a given industry have nothing to do with true joint products or by-products.

The presence of these products in the make matrix cannot but have a perturbative influence on the calculations which can be made. The UN Handbook of Input-Output Table indicates indeed that to adhere to the classification of products it is necessary to separate secondary products or joint products from the main products of an industry. Several methods can be used for the treatment of these secondary products resulting from production technology: the negative transfer method, the aggregation or positive transfer method, and the transfer of outputs and inputs. But none of them is fully satisfactory, and sometimes they can involve negative values, which may arise for purely statistical reasons.

These explanations about the statistical field shed some light on the complexity of compiling data on production activities, and bring us finally to the theoretical world, which is Sraffa's and Morishima's world, and the world we are concerned with. Here there is no particular constraint on the maximum number of industries or commodities that can be dealt with. After showing in the next two sections why Sraffa's as well as Morishima' theories of joint production are wrong, we shall go back to the labor theory of value, demonstrating that values are not incompatible with joint production, and also that even when there is joint production, there is no need to start with square matrices.

**4. Sraffa's theory of joint production**

There are two important things to note as regards the way Sraffa deals with joint production, which he starts to do in chapter VII of "Production of commodities"...: the first one regards his definition of joint production, which is right, and the second one concerns the way he tries to solve his system of equations, which is wrong and has nothing to do with values.

Let us underline first that in the universe of theory we can get rid of absolutely all of the secondary products, because there is no statistical limitation regarding the number of industries that we can accommodate. Therefore an "industry" can be defined much more
precisely than in the field of statistics: instead of regrouping establishments where "only a single, or (non-ancillary) productive activity ... accounts for **most** of the value added", which implies *a contrario* the existence of secondary activities and secondary products, nothing prevents us from deciding that an industry regroups all the establishments where a single productive activity accounts for *all* the value added. In fact it comes down to considering that at a first level of the theory there is a complete correspondence between each commodity and each industry, which is by the way the assumption that has prevailed so far, and exactly the same as Sraffa's and Morishima's assumption before they introduce joint production.

In passing these observations clearly show that the rather vague notion of industry, in the common sense that it has in the real world<sup>7</sup>, is quite different from the statistical concept of industry, which is itself different from the theoretical concept of industry such as it can be defined by economists. It follows that the concept of industry used by statisticians should not be confused with the concept of industry used by theoreticians. It is an interesting epistemological finding.

As far as we are concerned, in adopting a quite restrictive concept of industry corresponding to a particular technical process we should also be aware that, since there are more establishments and therefore more producers than industries which by definition regroup them, we always deal (and any theory does so) with averages. Indeed the techniques and methods of production generally differ - even if it is sometimes very slightly, from one producer to another, for obvious reasons of differences in location, productivity, types of machines, etc.

Therefore the processes for producing the same commodity are never exactly the same for all producers. If we are at the level of the production of one commodity in the whole economy, it means that the conditions of production and the production process for a given commodity correspond necessarily to the average process for this commodity at a given time, with such an average spanning many establishments and many production processes. At this stage to each industry producing a particular commodity corresponds therefore a single (and average) production process.

Before introducing joint production in his theory, and as we already indicated, Sraffa is thus right in deciding that the elementary unit of production is an industry identified to a single process of production. The question of the validity of the labor theory of value in the case where there are several techniques to produce the same commodity has been raised, but Toker has shown quite convincingly, in a note on the ""negative' quantities of embodied labor", published in 1984 in the Economic Journal, that such a situation is not a problem, as long as it is understood that what he calls (following Marx) the "market value"<sup>8</sup> of a commodity "is the weighted average of its individual values, weights being the market shares of the respective

<footnote n="7"><sup>7</sup>An industry is generally identified by broad categories of products, such as: *construction industry*, *chemical industry*, *petroleum industry*, *automotive industry*, *electronic industry*, *meatpacking industry*, and so on...</footnote>

<footnote n="8"><sup>8</sup>Since we are here at the stage of production, commodities have not yet been put on the market. Therefore it would be preferable to refer to the output proportions of the commodity in question, even though these proportions should be based on the market shares of the previous production period.</footnote>
techniques” (Toker, 1984, p. 152). From this observation it should be clear that each industry represents an average process of production, as long as joint production does not come into the picture.

After having rightly identified an industry with a single (and therefore an average) production process Sraffa then gets rid of the assumption that each commodity is produced by a separate industry, to suppose instead that a separate industry, and therefore a single process, can jointly produce two different commodities. From this it is clear that joint production as he defines it is identified to the production of exclusive by-products, which have nothing to do with secondary products that are produced by different processes. In such a case (a single process producing two different commodities) Sraffa explains rightly that there will be a single equation for the determination of two prices, and if the situation is generalized, more prices overall than equations, which is not sufficient to determine these prices.

Sraffa sees also clearly what constitutes a correct solution to overcome this difficulty: "In these circumstances there will be room for a second parallel process which will produce the two commodities by a different method and, as we shall suppose first, in different proportions. Such a parallel process will not only be possible, it will be necessary if the number of processes is to be brought to equality with the number of commodities so that the prices may be determined. We shall therefore go one step further and assume that in such cases a second process or industry does in fact exist" (Sraffa, 1960, p. 51).

Let us stress the fact that this assumption is a perfectly legitimate one, because in the real world, as we already noted, there are always a number of producers, and as already discussed there is no particular reason why it should not be the case for processes (or industries) producing joint products. There is also no reason for these processes to be strictly identical: on the contrary there is every reason to think that different producers will use production processes which are different, even though the differences are not very important. It is also quite probable that these producers will not produce the joint products (by-products) exactly in the same proportions.

In other words, since industries are identified with processes and since each one results from the aggregation of a number of producers and processes, there is no theoretical difficulty in doing things the other way around and disaggregating a given industry in as many processes (and industries) as the number of by-products that it produces.

Sraffa is therefore right when he generalizes his position in saying that the same result, i.e. the possibility to determine prices, would be achieved "provided that the number of independent processes in the system was equal to the number of commodities produced" (Sraffa, 1960, p.52), and by considering "a system of k distinct processes each of which turns out, in various proportions, the same k products" (Ibidem). It follows that "an industry or production-process is consequently characterized, no longer by the commodity which it produces, but by the proportions in which it uses and the proportions in which it produces, the various commodities" (Sraffa, 1960, P. 53).