On this basis, Sraffa's joint production equations present themselves as follows:

$$ 
\begin{align*}
(A_1 p_a + B_1 p_b + \dots + K_1 p_k)(1+r) + Lw &= A_{(1)} p_a + B_{(1)} p_b + \dots + K_{(1)} p_k \\
(A_2 p_a + B_2 p_b + \dots + K_2 p_k)(1+r) + Lw &= A_{(2)} p_a + B_{(2)} p_b + \dots + K_{(2)} p_k \\
\dots& \\
(A_k p_a + B_k p_b + \dots + K_k p_k)(1+r) + Lw &= A_{(k)} p_a + B_{(k)} p_b + \dots + K_{(k)} p_k
\end{align*}
$$

It is nevertheless at this juncture that problems start to arise. Indeed, to be able to deal with changes in the distribution (in w and r), prices and wages must be necessarily expressed in terms of the Standard commodity, the wage being defined as a share of the Standard net product, which Sraffa actually intends to construct in the case of joint production in chapter VIII of his book. In order to do so it is necessary for him to transform the above equations, through the definition of ad hoc multipliers, in such a way that products will appear in the same proportions on the left and right sides of these equations. This is a condition for the definition of R, the Standard ratio (which is equal to the maximum rate of profits when we have $w = 0$).

However, in this new context, some products may appear on the right side of the equations, as joint products, and not on the left side, as means of productions. Since these products cannot for this reason be part of the Standard commodity, they have to be eliminated, which implies the occurrence of negative multipliers and therefore of negative quantities in the Standard commodity. This does not seem to bother Sraffa, who writes that in the case of the Standard commodity "there is fortunately no insuperable difficulty in conceiving as real the negative quantities that are liable to occur among its components; these can be interpreted, by analogy with the accounting concept, as liabilities or debts, while the positive components will be regarded as assets" (Sraffa, 1960, pp. 56-57).

This quote shows that here we take another step further towards the realm of nonsense, because wages are defined in terms of the Standard commodity, like in fact all prices, which have necessarily the same dimension as the Standard, i.e. as their measurement unit, and wages are nothing else than a share of the net Standard product. It means therefore that with this new definition and construction of the Standard commodity, in Sraffa's theoretical universe workers would be paid in units of this strange thing, made not of final goods, i.e. consumption goods, but of intermediate commodities, and partly of debts and liabilities!

In the same vein, in the same chapter VIII of "Production of commodities", devoted to "the Standard System with Joint Products", Sraffa goes as far as explaining that: "Thus a Standard commodity which includes both positive and negative quantities can be adopted as money of account without too great a stretch of imagination provided that the unit is conceived as representing, like a share in a company, a fraction of each asset and of each liability, the latter as in the shape of an obligation to deliver without payment certain quantities of particular commodities" (Sraffa, 1960, p. 57). One must nevertheless acknowledge that such a view corresponds to a very strange conception of a money of account, which in fact borders to the grotesque.
When a theory thus leads to strange results, and in particular to totally unrealistic ones, like it
is the case for Sraffa's theory of joint production, one could think that the theory would be
abandoned or changed to produce a new or modified theory which would be able to generate
more realistic results. But it is not the path followed by Sraffa, nor by his followers, who seem
to think that since his theory is fine, the real world, and in particular the nature of wages and
money, can be skewed is such a way that it would ultimately fit to the theory. This is however
a strange epistemological attitude, to say the least.

But it is not Morishima's conception, since he decides on the contrary to modify the theory,
but in so doing thinks that he can also throw the baby with the bath water, and therefore
jettison the labor theory of value. This leads us to discuss now his conception, or rather his
misconception, of the labor theory of value, because it is based on the failure of Sraffa's
theory properly to integrate joint production.

**5. Morishima's misconception of the labor theory of value**

With the theory of joint production it is an important part of the theory of production prices
which produces the bizarre outcomes that we just described, and which lead us quite far from
the real world. One should normally consider that this theory is thus refuted and then
conclude that there is something wrong at least with this part of the theory. On this
background, the next step should be that this particular part of the theory should be
abandoned, in order to be reworked or rethought on different bases. But for Sraffa this was
impossible, because joint production is just an introduction and an indispensable element of
his theory of fixed capital, which follows in the next chapter of his book. Owing to these
shaky bases, it is no surprise that we already showed, in chapter 4, that this last one was also
deeply flawed.

Quite differently from Sraffa, Morishima does recognize the absurdity of negative quantities,
and he is convinced that he can remedy these difficulties by developing an alternative theory,
based on the Von Neumann's model. But what is not understandable is why he considers at
the same time that the theoretical flaws of the theory of production prices invalidate the labor
theory of value, which has not at all the same conceptual background, in particular as regards
its Standard, which as we demonstrated has nothing to do with any kind of composite
commodity!

For Morishima this question of negative quantities of commodities appearing in Sraffa's
theory of joint production is indeed considered as the crux of the matter, and as the basis for
his strong criticism of the labor theory of value. Following Sraffa, he starts by considering
that negative quantities of commodities appear in the theory of production prices. At the same
time he believes that this theory can define the value of commodities as made of "embodied
labor". For him it means ipso facto that there are negative labor values! But the values as he
defines them, i.e. through the same kind of equations as the ones above, are nothing more than
production prices when the rate of profits is equal to zero, which explains why he is wrong,
and this on several grounds. Then we are no longer in a problematics of transformation of
values into prices, but on the contrary in a reverse logic of transformation of prices into
values, which is another kettle of fish!

First it must be recalled that labor values are defined at the level of production, before any
sale has taken place, which implies that no profit has been obtained yet, whereas production
prices are defined at the level of distribution, because they incorporate not only wages but
also a uniform rate of profits, which implies (even if the rate of profit is equal to zero), that
prices have been realized by the sale of commodities on the market.

Moreover, as we recalled above, the dimension of prices in this theory is the dimension of
their measurement unit, i.e. the Standard commodity, which is a composite and heterogeneous
commodity. It is indeed a necessary condition for the existence of a linear relationship
between wages and the rate of profits. In passing, all these particularities show that to speak
of production prices is quite inappropriate, and that it would be much better and accurate to
name them distribution prices. It remains that these prices have nothing to do with time,
which is the dimension of values.

Morishima wants nevertheless to demonstrate that the labor theory of value, like Sraffa's
theory of production prices (of which he considers that the labor theory of value is a particular
case!) fails when it comes to joint production. This is the reason why, at the beginning of the
final chapter of his book, he tries to show it by giving an example of negative labor values.
But the big problem which arises then is that his example is completely absurd! (Morishima,
1973, Chapter 14, pp. 181-182).

To show it, let us present Morishima's example with a system of two matrices, one use matrix
of inputs and one make matrix of outputs. It replaces the simple Leontief model, in which one
industry produces only one commodity, and each commodity is produced only by one
industry. Leontief model is considered as a symmetric model, with therefore only one square
matrix, which cannot handle joint production, because there is no distinction between
commodities and industries.

The make-use model was introduced by the United Nations in the 1968 System of National
Accounts, and is well explained in a paper published in 2002 by Guo, Lawson and Planting,
from the US Bureau of Economic Analysis: "From Make-Use to Symmetric I-O Tables: an
Assessment of Alternative Technology Assumptions". We can therefore insert Morishima's
data into such a model, which is done in table 8.1.
<table><caption>Table 8.1 - Morishima's example of joint production presented in a make-use table</caption><thead><tr><th rowspan="2"></th><th colspan="3">Commodities Outputs</th><th colspan="3">Industries (processes)</th><th rowspan="2">Total Output</th></tr><tr><th>1</th><th>2</th><th>3</th><th>1</th><th>2</th><th>3</th></tr></thead><tbody><tr><td>Commodities Inputs</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>1 - Intermediate good</td><td></td><td></td><td></td><td>0.7</td><td>0.9</td><td>0.9</td><td>2.0</td></tr><tr><td>2 - New fixed capital good</td><td></td><td></td><td></td><td>0.5</td><td>0</td><td>0</td><td>1</td></tr><tr><td>3 - Old fixed capital good</td><td></td><td></td><td></td><td>0</td><td>0.5</td><td>0</td><td>0.5</td></tr><tr><td>Industries (processes)</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>1</td><td>1.0</td><td>0</td><td>0.5</td><td></td><td></td><td></td><td></td></tr><tr><td>2</td><td>1.0</td><td>0</td><td>0</td><td></td><td></td><td></td><td></td></tr><tr><td>3</td><td>0</td><td>1</td><td>0</td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td colspan="3">Labor</td><td></td></tr><tr><td>Value added</td><td>- 0.5</td><td>0.5</td><td>0</td><td>1</td><td>1</td><td>1</td><td></td></tr><tr><td>Total Input</td><td>2.5</td><td>0.5</td><td>0.5</td><td></td><td></td><td></td><td></td></tr></tbody></table>

We can see from this table 8.1 that in Morishima's example there is only one intermediate commodity, named good 1, and one capital good, which can serve for two periods. The new fixed capital good is designated as good 2 and the old fixed capital good is designated as good 3. The fixed capital goods are not used to product themselves but to produce the intermediate commodity, which can thus be produced by two industries (or processes), 1 and 2, utilizing the new and old fixed capital goods, respectively. Like in Sraffa's treatment of fixed capital, the old capital good is a by-product of process 1 using the new capital good. The industry (or process) which produces the new fixed capital good is called process 3.

Input-output coefficients are given in the table, where Morishima assumes that process 2 requires a greater amount of the circulating capital good (i.e. 0.9) than process 1 (i.e. 0.7), because the former uses the old fixed capital good and the latter the new one. Then the so-called “values” are calculated from the following equations, each corresponding to an industry, where inputs are on the left side, and outputs on the right side:

$$0.7\lambda_1 + 0.5\lambda_2 + \dots + 1 = \lambda_1 + \dots + 0.5\lambda_3$$

$$0.9\lambda_1 + \dots + 0.5\lambda_3 + 1 = \lambda_1 + \dots + \dots$$

$$0.9\lambda_1 + \dots + \lambda_3 + 1 = \dots + \lambda_2 + \dots$$

Solving these equations, Morishima obtains negative values: $\lambda_1 = -50$, $\lambda_2 = -44$, $\lambda_3 = -12$. But are this *ad hoc* system and its results significant? Obviously the answer is no, because its assumptions are incompatible with common sense, i.e. the sustainability and therefore the mere existence of the system, and this because of two errors:
- First the combined inputs of the intermediate good (i.e. 2.5) appearing in the last row of the table, are larger than the production of the same good (i.e. 2), which is incompatible with a self-replacing state;

- Second, because the treatment of fixed capital does not correspond either to Sraffa's treatment of fixed capital – however flawed it may be (Sraffa, 1960, pp. 76-80), or to a self-replacing state. Since for Morishima as well as Sraffa, fixed capital transfers its value to the product, then if there is one unit of new fixed capital good 2 produced at each period by process 3 and serving for two periods, which is the case in Morishima's example, then at the beginning of each period and therefore during each period this same unit of new fixed capital good 2 should be used as an input, in whatever process, with half of the value of this new fixed capital good transferred to the product, and the other half transformed into the old fixed capital good 3, as a by-product of the processes where it has been used (in this example, process 1 only). Otherwise fixed capital does not transfer its value to the product, which is what happens in process 1 (even though it is wrong). It is the reason why there is a net input of 0.5 for the new fixed capital good in the above table, which is not compatible with a self-replacing state. However the treatment of the half-unit of old fixed capital good is correct, because at each period its full (residual) value of 0.5 is transferred to the product by industry 2, then without any by-product.

However Morishima acknowledges that “the case of all goods having negative values is obtained only when the system does not satisfy the conditions of productiveness.” (Morishima, 1973, p. 182). In this example the corresponding input-coefficient matrix is not productive because the system uses more input of intermediate good 1 (2.5 units in total) than its produced quantity (only 2 units). One can therefore wonder why Morishima bothered to give an example which precisely did not meet these conditions, and has therefore nothing to do with the real world.

Then Morishima transforms his example by merging the two first processes producing good 1, and keeping only their average, which gives him the following system of equations, which he calls the neo-classical system:

$$0.8\lambda_1 + 0.25\lambda_2 + 1 = \lambda_1$$

$$0.9\lambda_1 + \dots + 1 = \lambda_2$$

He notes that we obtain again the same negative values of $\lambda_1 = -50$, $\lambda_2 = -44$, which is not surprising because the first equation is the average of the former two first ones: indeed this does not prevent the matrix from being non-productive, since there is 1.7 units of good 1 used as inputs whereas as it is easy to see only 1 unit is produced. He concludes from this second system that “the productiveness of the neo-classical system is necessary and sufficient for the positivity of the values of the circulating and new fixed capital good”.

Finally from there he develops a last case where the input-coefficient matrix is productive, and which is the following:
$$ \left. \begin{array}{l} 0.7\lambda_1 + 0.5\lambda_2 + \dots + 1 = \lambda_1 + \dots + 0.5\lambda_3 \\ 0.9\lambda_1 + \dots + 0.5\lambda_3 + 1 = \lambda_1 + \dots + \dots \\ 0.2\lambda_1 + \dots + \dots + 0.5 = \dots + \lambda_2 + \dots \end{array} \right\} \text{Morishima's system of equations} $$

He notes however that despite the fact that we obtain positive values for $\lambda_1$ and $\lambda_2$, at 7.5 and 2.0 respectively, negative values can still appear, which is the case for the old fixed capital good 3, its value being $\lambda_3 = -0.5$.

In fact it is so because Morishima does not fully realize the consequences of his own assumption that there is only one fixed capital good, which has a two period life and thus appears with two ages in the production processes: age 0, when it is produced by process 3 and used by process 1, and age 1 when it is used by process 2. To be sure, this does not prevent from naming the same goods of different ages goods 2 and 3, like he does, but it is not a pure convention, because there is a link between goods 2 and 3: indeed on the basis of Sraffa's and Morishima's assumptions regarding fixed capital and the transmission of its value to the product, good 3 is nothing more than good 2 which has lost, because it is one year old, a part of its value corresponding to its amortization in the process where it is used.

This forbids absolutely to write the first of the above three equations like Morishima does. In fact, to rewrite it properly, we must first take into account that fixed capital good 2 transfers in process 1 a value corresponding to its amortization, with its residual value, which Morishima seems to set at half its original value, appearing as a by-product in the form of good 3. We could be tempted to write it as in the following equation:

$$ 0.7\lambda_1 + 0.5\lambda_2 + \dots + 1 = \lambda_1 + \dots + (1-0.5)\lambda_2 $$

But we can see that both terms in $\lambda_2$ cancel, which precludes any transfer of value to the product from the new fixed capital good 2. As a consequence, if there must be any transfer of value to $\lambda_1$ (the value of the intermediate good) the equation must be rather written as:

$$ 0.7\lambda_1 + (1-0.5)\lambda_2 + \dots + 1 = \lambda_1, \text{ which gives us:} $$

$$ 0.7\lambda_1 + \lambda_2 + \dots + 1 = \lambda_1 + \dots + 0.5\lambda_2 $$

If we then consider like our author that the residual value of good 2 appears as a by-product, under the form of a distinct good which is fixed capital good 3, it is not 0.5 units, but one full unit of this new good 3 which must necessarily appear on the right side of the equation:

$$ 0.7\lambda_1 + \lambda_2 + \dots + 1 = \lambda_1 + \dots + \lambda_3 $$

The link between the two equations is obviously that $\lambda_3 = (1-0.5)\lambda_2$, assuming a 50% amortization of good 2 in the first period, which is a simple linear amortization rule (meaning that a fixed capital good is amortized in as many equal shares as the number of its periods of use), if we compare it to Sraffa's complicated demonstration, but corresponds to the fact that
here there is no rate of profit to complicate the matter. But then in the second equation $0.5 \lambda_3$ should in turn be replaced by $\lambda_3$, and we would therefore obtain quite a different - but coherent - system, which would be :

$$ \left. \begin{array}{l} 0.7\lambda_1 + \lambda_2 + \cdots + 1 = \lambda_1 + \cdots + \lambda_3 \\ 0.9\lambda_1 + \cdots + \lambda_3 + \cdots + 1 = \lambda_1 + \cdots + \cdots \\ 0.2\lambda_1 + \cdots + \cdots + 0.5 = \cdots + \lambda_2 + \cdots \end{array} \right\} \text{Correct system of equations} $$

However the two first equations are not independent, because of the amortization rule which links $\lambda_3$ to $\lambda_2$ (here it is $\lambda_3 = (1-0.5)\lambda_2$), and would make the system overdetermined, with 4 equations and 3 unknowns. The only way to avoid that is to sum the first two equations, to obtain finally the following system, where we can see that $\lambda_3$ has been eliminated:

$$ 1.6\lambda_1 + \lambda_2 + \cdots + 2 = 2\lambda_1 \quad | \quad (1) $$

The elimination from the first equation of $\lambda_3$, i.e. the value of the by-product, confirms that it is not possible to define the value of a fixed capital good by treating it as a by-product, while treating it at the same time as an intermediate good which transfers its value through an amortization process, something which we demonstrated in a more general case in Section 4 of chapter 4 on the treatment of fixed capital in Sraffa's system.

In any case, this system (1) above has two positive solutions for values of goods 1 and 2, i.e. $\lambda_1 = 12.5$ and $\lambda_2 = 3$, with the amortization rule giving us the value of good 3, as $\lambda_3 = 1.5$.

From all the foregoing we cannot but infer that Morishima's ad hoc example is meaningless and quite flawed, and proves nothing. But our author does not realize it, and on the contrary uses this supposed inconsistency (coming in fact from his initial assumptions) to derive what he thinks are quite general rules dismissing the labor theory of value. Ultimately such an example justifies his attempt to build his case of replacing the labor value system by a Von Neumann model with inequations, from which are derived minimum labor requirements, rather than labor values. This transforms his system into a theory designed mainly for the choice of the most productive techniques, which has nothing to do with the real world. In this world indeed a number of techniques and processes exist at any point in time for the production of any commodity, and values as social values are necessarily an average of this multiple methods of production.

It seems nevertheless all the more useless to continue discussing at length Morishima's demonstration that it is in fact quite possible to show that joint production is perfectly compatible with the determination and calculation of non-negative labor values, which will be achieved in the next section.
**6. Obtaining positive values in a realistic model of joint production**

The difficulties that we encountered so far have shown clearly enough that joint production is
for economists a true problem and at the same time a tricky phenomenon, which raises a
number of question-marks about the best way to deal with it, in particular from a
mathematical point of view. Statisticians are generally considered as more familiar with
mathematics as many economists, and it should therefore be no surprise that the light at the
end of the tunnel of joint production came from them. Indeed, as soon as 1968, they proposed
two different methods for transferring secondary outputs and associated inputs by combining
the use and supply matrices mathematically in order to build symmetric (and therefore
invertible) input-output matrices. A quick presentation of these methods, and of the reason for
selecting that which seems the most appropriate for the resolution of our problem, will allow
us to propose a corresponding representation of a joint production system. This will in turn
provide a background for a mathematical determination of values in such a system.

**6.1. Statistical methods developed for dealing with joint production**

These methods are exposed in the United Nations 1999 “Handbook of Input-Output Table”,
already cited. The first method was based on what was called an “industry technology
assumption”, which “assumes that inputs are consumed in the same proportions by every
product produced by a given industry, which means that principal and secondary products are
all produced using the same technology, i.e. the same input structure” (United Nations, 1999,
p. 86). However this method was quickly found to break the fundamental economic rule that
products with different prices at a given moment (i.e. secondary products, as opposed to the
principal product of an industry) must reflect different costs or different technologies.
Therefore the UN statisticians preferred to recommend the use of another assumption, i.e. the
“commodity technology assumption”, which assumes that the input structure of the
technology that produces a given product is the same no matter where (by which industry) it
is produced. Although this second method is indeed better adapted to the secondary products
that statisticians have to deal with, “it tends to generate negative symmetric input-output
tables and requires the make and intermediate matrices of the use table to be squared” (United
Nations, 1999, p.87). This is the reason why this method is not widely used.

However, and as far as we are concerned, joint production is not defined in a way which applies to secondary products that appear in each industry (in the sense given to the word industry by statisticians), because these secondary products are produced with different costs, i.e. with different technologies. On the contrary our definition of joint production, given in section 1 of this chapter, precisely reduces the category of joint products to that of exclusive by-products, i.e. products that have exactly the same input structure. Therefore the criticism made to the industry technology assumption is not relevant in the case of exclusive by-products, and this assumption perfectly fits our definition of joint production.

This explains that we will adopt this industry technology assumption for our demonstration
regarding the determination of values within a system of joint production, and all the more so
that this assumption is also attractive for two reasons: first, it is applicable to the case of
rectangular input-output tables, but - and this is the second and most important reason, the
method always generates positive symmetric input-output tables, which can be represented by
square matrices.

**6.2. Joint production as a physical process**

Before developing a mathematical solution, we can summarize the new view of production
that emerges when we take into account joint production, by redrawing the scheme that we
used (in section 2 of chapter 5) to describe production as a physical process. Let us stress that
this scheme is not a matrix, and that each line represents an industry (as a distinct technical
process). The *i* and *j* indices refer to commodities and industries, respectively. The situation
prevailing previously, such as represented in the scheme on page 102, was rather simple,
because to each commodity only one industry was corresponding, and thus the distinction
made between industries was the same as the distinction between commodities, which were
separated between three categories: intermediate commodities of the first type (listed from 1
to *k*), of the second type (listed from *k*+1 to *n*), and final goods (listed from *n* + 1 to *s*).

But now, in the more complicated world of joint production, the one-to-one correspondence
between commodities and industries (or processes) has disappeared. To be sure the same
distinction between various categories of commodities still exists, but we cannot use it in the
same way to establish an equivalent distinction between industries. Indeed we can no more
distinguish between those producing intermediate commodities of the first type, of the second
type, and final commodities, because they can produce in their outputs several commodities
belonging theoretically to anyone of these three categories. If we still can distinguish between
industries, it is no more on the basis of their outputs, but only on the basis of the commodities
that they use as their inputs: some will use only intermediate commodities of the first type,
and others will use both categories of intermediate commodities. Thus there is no more any
reason why the number of industries should coincide with the number of commodities.
Therefore the industries (or processes) using only intermediate commodities of the first type
will be listed from 1 to *l*, the industries using both types of intermediate commodities will be
listed from *l*+1 to *q* and the industries producing final commodities from *q*+1 to *t*.
Commodities will continue to be listed with the same indices as before.

As for the by-products of each industry (or process), most authors having worked on the
question, starting with Sraffa, seem to consider that in the area of joint production any
industry can produce any kind of commodity, belonging to any of the above categories, or at
least such an assumption is implicit in their models, since the question is never really
discussed. As far as we are concerned, and on the basis of the very definition of joint products
as exclusive by-products having exactly the same structure of inputs, it seems nevertheless
extremely doubtful that whatever industry should product jointly as by-products (distinct from
secondary products) both intermediate commodities and final commodities, for a first and
simple reason which has to do with the differentiation of final commodities.

This means that from one final commodity to the other, and even though two commodities are
almost close substitutes, there are differences at the end of the supply chain which imply at
the very least some differences in the structure of inputs that have been transformed to
produce them. And we already noted that the slightest difference in the input structure of two
commodities is sufficient to prevent them from being exclusive by-products. This is true for
consumption goods as well as for these other final goods that are machines or more generally
fixed capital goods. In fact by-products are more prone to be found in the upstream part of the
production process, i.e. at the stage of production of primary commodities or commodities
immediately derived from them, which are also by definition intermediate commodities.

Another and mutually reinforcing reason why by-products of these two different types seem
quite improbable is that intermediate commodities are sold by producers only to other
producers, and never to consumers. It implies that final goods sold to consumers reach them
through distinct and separate distribution and marketing channels that increase or create their
differentiation, since they change their cost and input structure. To be sure machines of fixed
capital goods are also sold from producers to producers, but as we just noted it would be quite
difficult to find out any plausible example of a fixed capital good as a by-product.

All these remarks obviously have a bearing on the possible representations of joint production
with exclusive by-products. If we focus first on industries (as processes) whose only inputs
are intermediate commodities of the first type (listed from 1 to *k*), we can reasonably assume
that there will be processes (listed from 1 to *l*) that produce by-products of the same type
(listed from 1 to *k*), and even simultaneously by-products that are intermediate commodities
of the second type (listed from *k*+1 to *n*). But it seems highly improbable that such processes
produce simultaneously final goods (listed from *n*+1 to *s*) as by-products, for the reason just
developed, which explains that this possibility will therefore not be retained.

A second category of processes (listed from *l*+1 to *q*) includes those that use as inputs both
types of intermediate commodities, and since by definition intermediate commodities of the
second type do not enter in the production of intermediate commodities of the first type, the
by-products resulting from these processes will only be intermediate commodities of the
second type.

As for final commodities (listed from *n*+1 to *s*), even though it is difficult to imagine a
process which would produce intermediate goods of the first type and simultaneously a
consumption good or a fixed capital-good as by-products, one cannot *a contrario* preclude
that some particular processes (listed from *q*+1 to *t*) might produce simultaneously
intermediate goods of the second type and such final goods (as an example one can think of
fruit-trees providing not only fruits – or final goods, but also some firewood – or intermediate
goods). Then the corresponding production system would have to be represented as in the
scheme appearing in table 8.2, where *Bij* corresponds to the quantity of product *i* used by
industry (or process) *j*, and *Dij* to the quantity of product *i* produced by industry (or process) *j*:
<table><caption>Table 8.2 - A schematic representation of a joint production system with by-products</caption><tbody><tr><td>B<sub>11</sub></td><td>..</td><td>B<sub>k1</sub></td><td>0</td><td>..</td><td>0</td><td>and</td><td>L<sub>1</sub></td><td>→</td><td>D<sub>11</sub></td><td>..</td><td>D<sub>k1</sub></td><td>D<sub>k+1,1</sub></td><td>...</td><td>D<sub>n1</sub></td><td>0</td><td>..</td><td>0</td></tr><tr><td>...</td><td>..</td><td>...</td><td>..</td><td>..</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>B<sub>1l</sub></td><td>..</td><td>B<sub>kl</sub></td><td>0</td><td>..</td><td>0</td><td>and</td><td>L<sub>l</sub></td><td>→</td><td>D<sub>1l</sub></td><td>..</td><td>D<sub>kl</sub></td><td>D<sub>k+1,l</sub></td><td>...</td><td>D<sub>nl</sub></td><td>0</td><td>..</td><td>0</td></tr><tr><td>B<sub>1,l+1</sub></td><td>..</td><td>B<sub>k,l+1</sub></td><td>B<sub>k+1,l+1</sub></td><td>..</td><td>B<sub>n,l+1</sub></td><td>and</td><td>L<sub>l+1</sub></td><td>→</td><td>0</td><td>..</td><td>0</td><td>D<sub>k+1,l+1</sub></td><td>...</td><td>D<sub>n,l+1</sub></td><td>0</td><td>..</td><td>0</td></tr><tr><td>...</td><td>..</td><td>...</td><td>...</td><td>..</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>B<sub>1q</sub></td><td>..</td><td>B<sub>kq</sub></td><td>B<sub>k+1,q</sub></td><td>..</td><td>B<sub>nq</sub></td><td>and</td><td>L<sub>q</sub></td><td>→</td><td>0</td><td>..</td><td>0</td><td>D<sub>k+1,q</sub></td><td>...</td><td>D<sub>nq</sub></td><td>0</td><td>..</td><td>0</td></tr><tr><td>B<sub>1,q+1</sub></td><td>..</td><td>B<sub>k,q+1</sub></td><td>B<sub>k+1,q+1</sub></td><td>..</td><td>B<sub>n,q</sub></td><td>and</td><td>L<sub>q+1</sub></td><td>→</td><td>0</td><td>..</td><td>0</td><td>D<sub>k+1,q+1</sub></td><td>...</td><td>D<sub>n,q+1</sub></td><td>D<sub>n+1,q+1</sub></td><td>...</td><td>D<sub>s,q+1</sub></td></tr><tr><td>...</td><td>..</td><td>...</td><td>...</td><td>..</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>B<sub>1t</sub></td><td>..</td><td>B<sub>kt</sub></td><td>B<sub>k+1,t</sub></td><td>..</td><td>B<sub>nt</sub></td><td>and</td><td>L<sub>t</sub></td><td>→</td><td>0</td><td>..</td><td>0</td><td>D<sub>k+1,t</sub></td><td>...</td><td>D<sub>nt</sub></td><td>D<sub>n+1,t</sub></td><td>...</td><td>D<sub>st</sub></td></tr></tbody></table>

This table 8.2 is built on the assumption that despite the existence of exclusive by-products, which might potentially imply that any industry could simultaneously produce any kind of goods, be they intermediate or final, there are nevertheless various types of industries, depending on the nature of commodities that they produce. This seems indeed to correspond more closely to the real world. It means that industries noted from 1 to $l$ use only basic goods of the first type to produce as exclusive by-products both basic goods of the first type, noted from 1 to $k$, and of the second type, noted from $k+1$ to $n$. Industries noted from $l+1$ to $q$ use both types of basic commodities to produce as exclusive by-products basic goods of the second type only, noted from $k+1$ to $n$. Finally industries noted from $q+1$ to $t$ use both types of basic commodities to produce as exclusive by-products both basic goods of the second type, noted from $k+1$ to $n$ and final goods noted from $n+1$ to $s$.

It must also be emphasized that by-products, despite the attention devoted to them, are not so frequent in the real world, and that when they occur there are rarely more than two by-products produced by the same process. As a consequence, even in the areas (in the above table) where some outputs are deemed potentially to be found, most of the quantities in the cells could well be zero. But this should not jeopardize the mathematical representation of the system.

Coming back to the industry technology assumption, let us recall that its basic assumption is that inputs are consumed in the same proportions by every commodity produced by a given industry, and thus in fact by every process, since to each process corresponds one industry. Let us stress again the fact that if such an assumption cannot be considered acceptable in the case of secondary commodities which are produced with different technologies, i.e. with different actual costs, by the same industry, it is nevertheless perfectly appropriate and acceptable in the case of pure or exclusive by-products, which are produced with exactly the same technology, and therefore the same proportions in inputs and the same costs.

On the particular point of the definition of an industry in a system of joint production, since a process can produce several by-products, and in order to have as many equations as unknowns, we must adopt the same assumption as Sraffa, who on this particular question was quite right : in order for the number of processes to be brought to equality with the number of commodities we must assume that each time an industry produces several by-products it can be decomposed (or disaggregated) in as many industries (or processes) as the number of by-
products that it produces. Alternatively, this same equality between the number of processes and by-products can also be achieved through another way, as Sraffa also rightly points out: “even if the two commodities were jointly produced by only *one* process, provided that they were *used* as means of production to produce a third commodity by two distinct processes; and, more generally, provided that the number of independent processes was equal to the number of commodities produced” (Sraffa, 1960, chapter 7, §50, p. 52).

From what we saw about processes and commodities, we can in any case consider this assumption as quite realistic, apart from the last corollary established by Sraffa, and according to which “*the number of independent processes was equal to the number of commodities produced*”. This equality is indeed a very particular case, which has no reason to exist in the general case and above all in the real world. If we recall that an industry is an intellectual construction, built through the compilation of data from many individual producers, each of them most often having its own process, each industry represents an average process or method of production. Therefore it is an assumption which is clearly neither unreasonable nor unrealistic to make: in the real world there are certainly more processes (or industries) than commodities, which obviously has an important consequence on mathematical representation of production processes, since it means that matrices will not be square, but rectangular.

### 6.3. A mathematical presentation of joint production

The starting point of this mathematical presentation is based on the make-use input-output tables developed by statisticians from the US Bureau of Economic Analysis (Guo, Lawson and Planting, 2002, pp. 1-42) and derived from the Input-Output Tables of the SNA as established by UN statisticians (United Nations, 1999, p. 88). To begin with the notations, we will keep the indices used so far, and therefore:

* *s* is the number of commodities (decomposed in *k*, *n*−*k* and *s*−*n*)

* *t* is the number of industries (decomposed in *l*, *q*−*l* and *t*−*q*)

* *U*, of dimensions (*s* × *t*) is the intermediate table of the use table (commodity by industry)

* *B*, of dimensions (*s* × *t*) is the use coefficient matrix (commodity by industry)

* *V*, of dimensions (*t* × *s*) is the make matrix (industry by commodity) transposed of the supply table *M* of dimensions (*s* × *t*) – commodity by industry, describing domestic production

* *D*, of dimensions (*t* × *s*) is the commodity output proportions matrix (industry by commodity)

* *g*<sub>*t*</sub> is the column vector of industry output

* *q*<sub>*s*</sub> is the column vector of commodity output

* *ĝ* is the diagonal matrix of industry output

* *q̂* is the diagonal matrix of commodity output
With these notations, the scheme of a joint production system as represented above in table 8.2 can be represented in table 8.3, where we have the two matrices U and V as defined above, and where the elements in capital letters represent quantities of the various commodities:

<table><caption>Table 8.3 A Make-Use Input-Output table for joint production</caption><thead><tr><th rowspan="2"></th><th colspan="3">Commodities</th><th colspan="3">Industries</th><th rowspan="2">Total<br>Output</th></tr><tr><th>l............k</th><th>k+1..................n</th><th>n+1..................s</th><th>l............l</th><th>l+1..................q</th><th>q+1..................t</th></tr></thead><tbody><tr><td>Com.</td><td></td><td></td><td></td><td colspan="3">Use Table: U</td><td></td></tr><tr><td>l</td><td></td><td></td><td></td><td>B11</td><td>B1,l+1</td><td>B1q</td><td>B1,q+1</td><td>B1t</td></tr><tr><td>...</td><td></td><td></td><td></td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>k</td><td></td><td></td><td></td><td>Bk1</td><td>Bk,l+1</td><td>Bkq</td><td>Bk,q+1</td><td>Bkt</td></tr><tr><td>k+1</td><td></td><td></td><td></td><td>0</td><td>0</td><td>Bk+1,l+1</td><td>Bk+1,q</td><td>Bk+1,q+1</td><td>Bk+1,t</td></tr><tr><td>...</td><td></td><td></td><td></td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>n</td><td></td><td></td><td></td><td>0</td><td>0</td><td>Bn,l+1</td><td>Bnq</td><td>Bn,q+1</td><td>Bnt</td></tr><tr><td>n+1</td><td></td><td></td><td></td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td></tr><tr><td>...</td><td></td><td></td><td></td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>s</td><td></td><td></td><td></td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td></tr><tr><td>Indus.</td><td colspan="3">Make Table: V</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>l</td><td>D11</td><td>...</td><td>Dk+1,l</td><td>Dn1</td><td>0</td><td>...</td><td>0</td><td></td></tr><tr><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td></td></tr><tr><td>l</td><td>D11</td><td>...</td><td>Dk+1,l</td><td>Dn1</td><td>0</td><td>...</td><td>0</td><td></td></tr><tr><td>l+1</td><td>0</td><td>...</td><td>Dk+1,l+1</td><td>Dn,l+1</td><td>0</td><td>...</td><td>0</td><td></td></tr><tr><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td></td></tr><tr><td>q</td><td>0</td><td>...</td><td>Dk+1,q</td><td>Dnq</td><td>0</td><td>...</td><td>0</td><td></td></tr><tr><td>q+1</td><td>0</td><td>...</td><td>Dk+1,q+1</td><td>Dn,q+1</td><td>Dn+1,q+1</td><td>Ds,q+1</td><td></td><td></td></tr><tr><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td></td></tr><tr><td>t</td><td>0</td><td>...</td><td>Dk+1,t</td><td>Dnt</td><td>Dn+1,t</td><td>Dst</td><td></td><td></td></tr><tr><td>Total<br>Input</td><td colspan="3">q'</td><td colspan="3">g'</td><td></td><td></td></tr></tbody></table>

Regarding now the notations, so far in this book we used to name $a_{ij}$ the production (or input-output) coefficients, which correspond to the share of the total quantity of a product $i$ used by an industry (or process) $j$.

Since one industry produced only one commodity, and reciprocally, these coefficients corresponded to a commodity-by-commodity square matrix, and we will keep them in this role. But the industry technology assumption, with which we are now working, allows us to start from a more general case of rectangular input-output tables, which are normally expected in the case where there are more industries ($l$, $q$ or $t$) than commodities ($k$, $n$ or $s$). As we shall see this will not prevent us from generating symmetric input-output tables.

Reserving thus the $a_{ij}$ notation to the particular case of a commodity-by-commodity square matrix, the data upon which we must rely are as usual the methods or processes of production,
represented by the share of each commodity's total output that is used as an input by each industry. We will rename these coefficients as $b_{ij}$, in the case of the use coefficient matrix (commodity by industry) derived from the use matrix $U$ in the above table, and which is therefore named $B$. This commodity-by-industry, direct requirements matrix $B$ has $s$ rows, corresponding to $s$ commodities, and $t$ columns, corresponding to $t$ industries (or processes), and is therefore a rectangular matrix of dimensions ($s$ x $t$). It shows (in rows) the share of each commodity's total output that is used as an input by each industry, which explains why the last rows from $n+1$ to $s$ are made of zeros, because final goods are not used as inputs. It also shows (in columns) the commodity composition of each industry's total inputs.

Matrix $B$ is derived by dividing the use matrix $U$ of dimensions ($s$ x $t$) by the vector of industry total output $g$ of dimensions ($t$ x $t$):

$$B = U\hat{g}^{-1} \qquad (1)$$

$$ \begin{array}{c} \text{Commodities} \\ \begin{bmatrix} 1 \\ \vdots \\ k \\ k+1 \\ \vdots \\ n \\ n+1 \\ \vdots \\ s \end{bmatrix} \end{array} \quad \begin{array}{c} \text{Industries} \\ \begin{bmatrix} 1 & \dots & l & \dots & l+1 & \dots & q & \dots & q+1 & \dots & t \\ b_{11} & b_{12} & b_{1,l+1} & b_{1q} & b_{1,q+1} & b_{1,t} \\ b_{k1} & b_{kl} & b_{k,l+1} & b_{kq} & b_{k,q+1} & b_{kt} \\ 0 & 0 & b_{k+1,l+1} & b_{k+1,q} & b_{k+1,q+1} & b_{k+1,t} \\ 0 & 0 & b_{n,l+1} & b_{nq} & b_{n,q+1} & b_{nt} \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{bmatrix} \end{array} $$

(2)

In the case of joint production, one particular process can produce several different products. Even though in the real world pure or exclusive by-products are quite rare, and there are not so often more than two exclusive by-products for a particular joint production process, from a theoretical point of view we must consider a general case in which any process is able to produce a variety of different joint-products, in the sense of exclusive by-products.

In order to give a mathematical representation of this reality, we will use below a "commodity-output-proportions" matrix $D$, which is an industry-by-commodity matrix. It is a matrix which has $t$ rows (the industries) and $s$ columns (the commodities), and is therefore also a rectangular matrix, of dimensions ($s$ x $t$). It shows (in rows) the commodity composition of each industry's total output of various commodities, which are exclusive by-products. It also shows (in columns) the share of each commodity's total output that is produced by each industry.
This “commodity-output-proportions” matrix $D$, which shows the shares of each commodity’s total output that are produced by each industry, is obtained by dividing the make matrix $V$ of dimensions ($t \times s$) by the vector of commodity output $q$ of dimensions ($s \times t$):

$$D = V\hat{q}^{-1} \qquad (3)$$

$$ \text{Industries} \qquad \left[ \begin{array}{c} l \\ \vdots \\ k \end{array} \right. \qquad \left[ \begin{array}{c} k+1 \\ \vdots \\ n \end{array} \right. \qquad \left. \begin{array}{c} n+1 \\ \vdots \\ s \end{array} \right] $$
$$ \begin{bmatrix} 1 \\ \vdots \\ l \\ l+1 \\ \vdots \\ q \\ q+1 \\ \vdots \\ t \end{bmatrix} \qquad \begin{bmatrix} d_{1l} & d_{kl} & d_{k+1,l} & d_{nl} & 0 & 0 \\ d_{1l} & d_{kl} & d_{k+1,l} & d_{nl} & 0 & 0 \\ 0 & 0 & d_{k+1,l+1} & d_{n,l+1} & 0 & 0 \\ 0 & 0 & d_{k+1,q} & d_{nq} & 0 & 0 \\ 0 & 0 & d_{k+1,q+1} & d_{n,q+1} & d_{n+1,q+1} & d_{s,q+1} \\ 0 & 0 & d_{k+1,t} & d_{nt} & d_{n+1,t} & d_{st} \end{bmatrix} \qquad (4) $$

On the basis of the industry technology assumption a product $j$ can be produced by a certain number of different industries $k$. Each industry $k$ needs $b_{ik}$ units of input $i$ per unit of industry product $j$, where $b_{ik}$, $i=1...n$ represents the industry technology of an industry $k$, and each industry $k$ produces only a part of the total output of product $j$. This proportion of industry $k$ in the total production of product $j$ has a notation $d_{kj}$. So all inputs $i$ needed to produce one unit of product $j$ by different industries can be written as follows:

$$a_{I,ij} = \sum_{k=1}^{n} b_{ik} d_{kj} \qquad (\text{where I refers to the industry technology}) \qquad (5)$$

The above formula shows that input $i$ required to produce one unit of product $j$ is a weighted average of the input structures of the industries where product $j$ is produced: the weights are the proportions $d_{kj}$ of each industry $k$ in the total production of product $j$.

To make things clearer, let us use words to develop the above equation for coefficient $a_{ij}$, i.e. the share of the total production of commodity $i$ which is used in the total production of commodity $j$:

$$a_{ij} = b_{i1}d_{j1} + b_{i2}d_{j2} + \dots + b_{ij}d_{jj} + \dots + b_{it}d_{jt} \qquad (6)$$

It means that $a_{ij}$ is the sum of the share of total production of good $i$ used in industry 1, multiplied by the share of industry 1 in the total production of good $j$, plus the share of total production of good $i$ used in industry 2, multiplied by the share of industry 2 in the total
production of good j, ... plus the share of total production of good i used in industry j, multiplied by the share of industry j in the total production of good j, ... plus the share of total production of good i used in industry t, multiplied by the share of industry t in the total production of good j.

Thus, through the additional information provided by the shares of various industries in the total production of a particular product (that could be defined as the “market shares” of these industries for this product), we are able to obtain the additional equations which allow for disentangling the initially hidden input structure of particular by-products. These additional equations in the form of equation (6) can be put in matrix format and thus be written as:

$$A_{I(s,s)} = B_{(s,t)} D_{(t,s)} \qquad (7)$$

As B is a commodity-by-industry rectangular matrix, of dimensions (s x t). D as defined above is an industry-by-commodity rectangular matrix, of dimensions (t x s). Therefore matrix $A_I$ is a commodity by commodity matrix of dimensions (s x s). Thus $A_I$ is the input-output coefficient matrix that describes commodities directly required to produce other commodities. It is a square matrix. For UN and US statisticians, in the sources referred to above, as well as for Peter Flaschel in a 1983 article on “Actual Labor Values in a General Model of Production” (Flaschel, 1983, pp. 435-454), which uses their method, this square matrix A is invertible. This is a necessity for statisticians in order to calculate the commodity-by-commodity, total requirements matrix (ITC), which is derived as:

$$T = [I - A]^{-1} = [I - BD]^{-1} \qquad (8)$$

But it is also a necessity for theoreticians, who want to calculate values by using the usual and well known formula:

$$V = VA + L$$

Which implies that $V = L(I - A)^{-1}$

One last difficulty deriving from the existence of joint production and which has to be addressed at this stage is indeed the nature of vector L, which appears in the above equation, because if we go back to our initial description of a joint production system in section 6.2. above, we note that the only information that we have regarding the quantities of labor time used in the system are the quantities by industry, which is logical because the nature of by-products does not allow to apportion these quantities to the various joint products of particular industry.

Therefore the row vector of these quantities of labor has for dimensions (1 x t) and is:

$$V^I = (l_1^I + l_2^I + \dots + l_t^I) \text{ where index } I \text{ stands for industry}$$
On this basis, if we wanted to calculate values, using matrix $B$, we would be unable to do it directly, because the unknown values to be determined would be both the values produced by each industry (vector $V^I$), and the values of each commodity, i.e. the elements of vector $V^C$:

$$V_{(1,t)}^I = V_{(1,s)}^C B_{(s,t)} + L_{(1,t)}^I \quad (9)$$

Fortunately, there is a solution to this problem, which comes down to using the same tool as the one devised by statisticians, i.e. matrix $D$ which gives us the commodity-output proportions. If indeed we multiply both terms of the above equation by matrix $D$, we obtain:

$$V_{(1,t)}^I D_{(t,s)} = V_{(1,s)}^C B_{(s,t)} D_{(t,s)} + L_{(1,t)}^I D_{(t,s)} \quad (10)$$

As was already demonstrated matrix $D$ indeed provides a key for passing from industries to commodities, not only through the transformation of matrix $B$, but also for the transformation of vector $V^I$ and $L^I$, since it allows us to write:

$$V^C = V^I D \quad \text{and} \quad L^C = L^I D, \text{ from which we derive:}$$

$$V_{(1,s)}^C = V_{(1,s)}^C A_{(s,s)} + L_{(1,s)}^C \quad (11)$$

Now that we have been able to overcome the difficulties linked to the existence of by-products, with the help of a method developed by statisticians and perfectly suited to the features of exclusive by-products, through the design of a matrix supposed to “well behave”, the calculation of actual labor values seems to be quite simple. From a theoretical point of view, however, things cannot be as simple as they seem, and this for several reasons.

The first reason for which things are not as simple as they might seem has to do with the basic data on which the method is based. Indeed, in its article just cited, Flaschel sticks fully to the SNA method, and since the quantities used by statisticians in their input-output tables are the monetary values of the commodities, he introduces prices in the calculation of the share of each industry in the production of joint products. This explains that the variables which appear in Flaschel's article are therefore what he calls the “relative sales values”, from which are derived the monetary market shares of these commodities. Prices are therefore introduced in the calculation of values, a position that Flaschel fully endorses when he writes: “for our intents, labor values $v$ were made to depend on prices $p$ by the use of sales values $C_{kj}$” (Flaschel, 1983, p. 453).

However such a position is wrong for various reasons. First, because to know the market shares implies to know which quantities have been sold on the market, and at which prices. But when we stand at the theoretical level of production and values, there are no prices, because as we already mentioned previously, values are logically and chronologically anterior to the sale of commodities at monetary prices, which appear only after production has taken place, and when this production is being sold on the market.
Another reason has also been emphasized by Toker, and has to do with the distinction between average values and individual values of commodities, which necessarily appear when the same commodity can be produced by several methods. This author indeed writes: "unlike single-production systems, in joint production systems prices at zero rate of profit are still prices *par excellence* but not labor values; in the latter perfect competition can ensure, at any rate of profit (including zero) the uniqueness of prices, but not the uniqueness of labor values" (Toker, 1984, p. 152)..

A last reason why Flaschel should not have based his calculation of labor values on monetary market shares is that values and prices do not have the same dimension: values have the dimension of time, which is the standard of values, and prices are pure dimensionless numbers, or scalars. Therefore they cannot be put at the same level, which means that values are not a special form of prices which would appear when the rate of profit is zero.

Moreover the use of monetary market shares is not needed as a mathematical necessity for the determination of values in the case of joint production, since matrix *D* above can perfectly be defined in terms of ratios between the quantities of a commodity produced in various industries and the total quantity of this same commodity produced by the whole production system: it is a matrix of “commodity-output proportions”. This is indeed what has been done above (p. 158) by writing: $D = Vq^{-1}$, with the elements of *V* being mere quantities.

There is also another reason for which things are not as simple as they might seem, which has to do with the existence of several distinct categories of commodities. Indeed, as soon as we introduce, as it is the case in this section, a distinction between final commodities and intermediate commodities, which is indispensable in a theory where production is seen as a transformation of the later into the former, things become more complicated than they are in the statistical world where this distinction is not taken into account. Then final goods are not transformed into intermediate goods, and there are two types of these last ones, which - as we already showed in previous chapters, has some consequences on the nature of matrices involved in the representation of production.

As a result matrix *A*, as the product of both matrices *B* and *D* [which themselves are as exposed above in (2) and (4)], presents itself in the following form, with matrices *B* and *D* decomposed into sub-matrices:

$$A = \begin{bmatrix} B_{11} & B_{12} & B_{13} \\ 0 & B_{22} & B_{23} \\ 0 & 0 & 0 \end{bmatrix} * \begin{bmatrix} D_{11} & D_{12} & 0 \\ 0 & D_{22} & 0 \\ 0 & D_{32} & D_{33} \end{bmatrix} = \begin{bmatrix} B_{11}D_{11} & B_{11}D_{12}+B_{12}D_{22}+B_{13}D_{32} & B_{13}D_{33} \\ 0 & B_{22}D_{22}+B_{23}D_{32} & B_{23}D_{33} \\ 0 & 0 & 0 \end{bmatrix} \quad (12)$$

It is obvious that the resulting matrix *A* is singular and therefore is not invertible, because matrix *B* is itself a singular matrix: its *s*-*n* last rows are zeros, which simply reflects the fact that final goods by definition do not enter into the production of any good. By getting rid of
these last rows, we can nevertheless rewrite matrix A as a new matrix $A^*$, which can be written as:

$$A^* = \begin{bmatrix} A_{11} & A_{12} & A_{13} \\ 0 & A_{22} & A_{23} \end{bmatrix} \qquad (13)$$

It is clear that $A^*$ is a rectangular matrix of dimensions $(n \times s)$, and therefore is not invertible.

As for the dimensions of the six submatrices contained in matrix $A^*$, they are the following:

$A_{11}$ is a square matrix with $k$ rows and $k$ columns, of dimensions $(k \times k)$

$A_{12}$ is a rectangular matrix with $k$ rows and $n-k$ columns, of dimensions $(k \times n-k)$

$A_{13}$ is a rectangular matrix with $k$ rows and $s-n$ columns, of dimensions $(k \times s-n)$

$A_{21}$ is a zero rectangular matrix, with $n-k$ rows and $k$ columns, of dimensions $(n-k \times k)$

$A_{22}$ is a square matrix with $n-k$ rows and $n-k$ columns, of dimensions $(n-k \times n-k)$

$A_{33}$ is a rectangular matrix with $n-k$ rows and $s-n$ columns, of dimensions $(n-k \times s-n)$

An observation which can be made from this matrix $A^*$ is that we can isolate from it another submatrix $A^#$, which is composed of the first two rows and first two columns of submatrices of matrix $A^*$, such as:

$$A^# = \begin{bmatrix} A_{11} & A_{12} \\ 0 & A_{22} \end{bmatrix} \qquad (14)$$

In fact submatrix $A_{11}$ gives us the technical coefficients that correspond to the share of intermediate goods of the first type which enter in their own production, submatrix $A_{12}$ those which correspond to the share of these same intermediate goods of the first type which enter in the production of intermediate goods of the second type, and matrix $A_{22}$ those which correspond to the share of these last goods which enter only in their own production.

To be sure $A^#$ is a square matrix of dimensions $n \times n$, but it columns are not linearly independent, which implies that $A^#$ is a singular matrix and therefore is not invertible. Like we already did for the calculation of values in the absence of joint production (see sub-section 2.2. of chapter 6), we must therefore solve the problem of value determination by using a step by step approach. Indeed, once the system is split into three subsystems, it becomes possible to represent it in the following matrix format, where we have three matrix equations:

$$V_1 = V_1 A_{11} + L_1 \qquad (15)$$

$$V_2 = V_1 A_{12} + V_2 A_{22} + L_2 \qquad (16)$$

$$V_3 = V_1 A_{13} + V_2 A_{23} + L_3 \qquad (17)$$

Only the first one is an independent matrix equation, in which $A_{11}$ is a $k$ by $k$ matrix with entries $a_{ij}$. Such a matrix corresponds to what is usually called an open Leontief model, i.e. to an economy where there is some external source of demand for each industry: here this
demand corresponds to the share of the production of intermediate commodities of the first type which is needed for the production of other intermediate and final commodities.

Matrix $A_{11}$ is positive ($A_{11} > 0$), because $a_{ij} \ge 0$ for all $i$ and $j$ and $a_{ij} > 0$ for some $i$ and $j$. We know also that $A_{11}$ has row sums not exceeding 1, meaning that:

$$ \sum_j a_{ij} \le 1 \text{ for all } i $$

Such a matrix is called sub-stochastic (it would be stochastic if each row sum were 1). In such a case the demonstration has been made by Peterson and Olinick that $(I - A_{11})^{-1} > 0$. To be sure, they make the demonstration for matrices where column sums (and not row sums) are not exceeding 1, because they use the transposed matrix of matrix $A_{11}$, but this does not change either the demonstration or its outcome.

Let us first recall that by definition, matrix $A$ is productive if there is a nonnegative vector $X$ such that $X \ge AX$. Their demonstration is then based on their theorem 5.1. (Peterson and Olinick, 1982, pp. 221-239):

**Theorem 5.1.:** A sub-stochastic matrix $A$ is productive if and only if $I - A$ is non-singular.

From theorem 5.1., matrix $I - A$ is invertible and necessarily then $(I - A_{11})^{-1} > 0$. The solution vector is $X = (I - A_{11})^{-1} B$.

This ensures that values are always positive, which allows us to write:

$$ V_1 = (I - A_{11})^{-1} L_1 \qquad (18) $$

Then, knowing $V_1$, the vector of values of intermediate commodities of the first type, from equation (4) we get $V_2$: the vector of values of intermediate commodities of the second type, and since $A_{22}$ is a square matrix of dimensions $n-k$ by $n-k$, we can solve the second equation:

$$ V_2 = V_1 A_{12} + V_2 A_{22} + L_2, \text{ by writing:} \qquad (19) $$

$$ V_2 = (I - A_{22})^{-1} (V_1 A_{12} + L_2) \qquad (20) $$

Finally, knowing $V_1$ and $V_2$, from equations (18) and (20) we obtain $V_3$, the vector of values of final commodities, without any other additional calculation than displayed by equation (17):

$$ V_3 = V_1 A_{13} + V_2 A_{23} + L_3 \qquad (21) $$

We thus demonstrated that values can be calculated in a coherent way in case of joint production. Thus we have arrived now at the end of this chapter on joint production, which
has shown that contrarily to Sraffa's and Morishima's assertions it is perfectly possible to
introduce joint production in a theoretical system without having negative labor values. In fact
such peculiarities appear in Sraffa's theory because of the need for him to build a Standard
commodity which appears to be quite strange in his system, and in Morishima's theory
because of the introduction of strange assumptions like the existence of non-productive
sectors that use some inputs in larger quantities than the produced quantities of these same
inputs.

This conclusion according to which there is no such thing as negative quantities of labor has
the merit to be in line with intuition, because in the real world, where indeed joint production
does exist, albeit in small proportions compared to overall production, this particularity is
never associated to negative values, being understood that, as Flaschel mentions it rightly:
"the labor value of a jointly produced free good should be zero" (Flaschel, 1983, p. 449)..

Although we disagree with Flaschel on a particular point, i.e. the need to use prices in the
calculation of values, we cannot but adhere to his conclusion which confirms the relevance of
the labor theory of value and the existence of "actual labor values in a general model of
production". What this chapter has shown indeed is that as long as it stays confined to the
theoretical sphere of production and is not unduly extended to that of exchange, Marx's labor
theory of value is perfectly compatible with joint production. On the basis of this finding we
can state a new general principle:

**Principle 15:** Joint production, defined as the production of several distinct commodities by a single method of production, is compatible with the existence and calculation of positive labor values, as long as different methods produce these commodities in different proportions.

All this allows us to come back in a concluding chapter of the second part of this book to the
concepts of production and value.

References

Flaschel, Peter (1983). *Actual Labor Values in a General Model of Production*. *Econometrica*, vol. 51, N°2 (March 1983), pp. 435-454.

Guo Jiemin, Lawson Ann, and Planting Mark (2002, November). *From Make-Use to Symmetric I-O Tables: an Assessment of Alternative Technology Assumptions*. Washington, DC: US Bureau of Economic Analysis. US Department of Commerce.

Horowitz, Karen J., and Planting, Mark (2006). *Concepts and Methods of the Input-Output Accounts*. Washington, DC: Bureau of Economic Analysis (September 2006, updated April 2009), US Department of Commerce.

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, sub-stochastic matrices, and positive solutions to matrix equations. *Mathematical modelling, 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.

Toker, Mehmet Ayhan (1984). A note on the ‘negative’ quantities of embodied labour. *The Economic Journal*, No 94 (March issue), pp. 149-154.

United Nations (1999). *Handbook of Input-Output Table. Compilation and Analysis*. Series F, N° 74. New-York: United Nations.

United Nations, OECD, IMF, World Bank, European Commission (2009). *System of National Accounts 2008*. United Nations document symbol: ST/ESA/STAT/SER.F/2/REV 5. New-York: United Nations.

United Nations - Department of Economic and Social Affairs (2014). *Studies in Methods. Handbook of National Accounting. Financial Production, Flows and Stocks in the System of national Accounts*. Series F, N° 113. New-York: United Nations.

United Nations - Department of Economic and Social Affairs (2015). *Central Product Classification (CPC)*. Version 2.1. Series M, N° 77. New-York: United Nations.
# Chapter 9. Conclusion on the concepts of production and value

The purpose of this short chapter is to recapitulate the principal lessons which have been drawn from the second part of this book on production and values. At the end of this part, it seems indeed useful to come back to these concepts, and to give more precisions about their content. We will come back to the principle that production is anterior to and as such independent from exchange, and that it never creates any surplus. We shall discuss the existence of values, as an operational concept, and their field of validity, and show that values are independent from the social structure of the productive system.

## 1. Production is independent from exchange

What this whole second part has shown is that to think the concept of production with as a point of departure the concept of transformation is perfectly possible and feasible, without encountering any contradiction. Moreover, this concept of production does not preclude at all to define values as the common property of commodities obtained through this transformation of other commodities. It is so because such a transformation always involves the intervention of human labor, which is the common point that makes them homogeneous, at a purely theoretical level, and therefore equivalent. And this indeed can be achieved, at least from a purely theoretical point of view, without any reference to what happens to these commodities after they have been produced, i.e. their exchange and distribution.

This approach is made possible because production cannot be defined otherwise than as a transformation, which ends up with the apparition of new material entities, thus called products or commodities. In other words, production is not an *ex nihilo* creation but a creation through the transformation of other and therefore pre-existing material entities.

As such, production is a much more general reality and concept than exchange. Production as transformation is first a natural and widespread phenomenon, which means that every existing thing in nature is the result of some prior transformation of other elements: this is the sense of Lavoisier's law, which implies that in the physical world there is no such thing as spontaneous generation. Production is an activity which takes place in the physical world and abides by Lavoisier's law, which means therefore that prior elements are themselves the result of a past transformation, and so on and so forth back to the primeval elements, i.e. the elements that appear in Mendeleev's table. We know now that these primeval elements were produced in the Big Bang and later in the heart of stars. Physical production has existed in nature for billions of years well before the existence of humanity.

To be sure economic production most often does not go so far as to use directly these primeval elements as the building blocks of its activity, but it uses raw materials, or primary commodities. Some of them are considered as “non-produced”, but in fact this only means that they have not been produced by man in a process that could be called “human production”, because they have been produced previously by nature itself. This is obvious in the case of all mineral primary commodities, like iron ore or oil. When we say that iron ore or
oil are produced, it means more precisely that they are extracted from their deposit and separated from other mineral constituents. In the case of iron ore mining or oil extraction, both activities must be seen as a physical transformation. In the case of oil, for instance, it took millions of years for nature to produce it through physical and chemical processes, meaning that human production by itself does not create crude oil, which already exists deep in underground deposits. Production of crude oil consists in multiple activities which start with exploration, continue with drilling and end up in pumping crude oil up (if needed) to surface.

In the case of agricultural products, things seem a little less obvious, because at first sight some of these products do not seem to preexist to human intervention. It should be clear however that the vast majority of agricultural product existed in nature before and without any human activity whatsoever. Grains of wheat or rice were produced by the corresponding grasses and could be gathered by men before these cereals started being cultivated by them, around 10 000 years ago. Game was hunted before animals were domesticated, and fruits were picked before the cultivation of fruit trees.

That this natural production seems to add something to the seeds which are sowed is also an illusion, because like minerals the products of the land are also the result of the transformation of pre-existing natural elements, through biological processes like photo-synthesis. Using solar energy transported by photons, such a process dissociates carbon dioxide from the air to obtain carbon atoms used by cellular machinery to create complex organic elements. The soil provides water brought by rain as well as nutrients like nitrogen, potassium and phosphorus. From a physical point of view, land is therefore nothing else than a substratum which stores all of these elements, including many organic compounds, as well as bacteria and fungi that often play a role in the production process of many plants. What human production does in the case of agricultural production is to organize, rationalize and accelerate these natural processes, thus increasing the quantities of commodities which are its outputs.

All these examples show well that production can be analyzed and understood as a concept without the existence of man or society, and therefore of any kind of exchange whatsoever between human beings.

## 2. Production never creates any surplus

These explanations make it easier to understand why human production, no more than natural production, never creates any surplus. This fact is certainly more straightforward in the case of mineral production, where the main elements which are going to be produced (for instance iron) already exist and are already present underground – albeit in another form (for instance hematite ore), before any human production takes place. In the case of iron, this human production consists in the extraction of this ore and in the separation of iron, first from the waste rock and second from its oxide, which is indeed a transformation (a change of form) and corresponds to our definition of production. But there is no material or physical difference, and hence no surplus, between the preexisting iron content of iron ore stored underground and the same iron once the iron ore treatment has been carried out.
Similarly to mineral production, it would be an abuse of language to consider that primary commodities coming from agricultural production constitute a surplus. As we already noted previously this is the result of wrongly considering that the seeds which are sowed are exactly identical to the grains which are harvested, which in all rigor is not true. This observation is still easier to understand in the case of fruit production, where the fruits which are picked on the trees are physically distinct from the seedlings which are planted to grow into fruit trees.

In both cases, what creates the illusion of a physical surplus is the existence of a number of elements (like hematite, or water from rain, carbon dioxide from the air, etc.) which are all going to be transformed into mineral or agricultural products (like iron or wheat), but pre-exist to any human activity, and in this case are therefore not the result of any human production. This is the reason why these elements may be considered as non-produced, which to some extent might look like another abuse of language, unless we precise that it means in fact that these elements are non-produced by human activity, i.e. by labor.

This finding allows us to understand that basic commodities in Sraffa's sense<fnref n="9" /> are made of raw materials (from mining or agricultural origin) and of all the commodities, mostly from industrial origin, that are used to produce these raw materials. Since it is obvious that non-basic commodities are themselves made of some raw materials that will therefore appear in the equations describing their production, it is clear that the corresponding share of these basic products will not appear simultaneously in the equations describing their own production. This phenomenon will also concern the commodities, other than raw materials, that are used in the production of basic goods. All this implies that, even though the existence of a surplus is impossible at the scale of the whole production system, a surplus necessarily appears in the sub-system describing the production of basic commodities. In turn this explains why the matrix representing this sub-system is productive and sub-stochastic.

As far as this notion of surplus is concerned, a last point must be emphasized, regarding the equations describing the system of production of basic commodities, such as they are written in Sraffa's "Production of commodities..." (Sraffa, 1960, § 2, p. 4). These equations are indeed misleading when they make the same raw material appear simultaneously on both the left and right hand sides of the same equation, because they tend to blur the fact that there is no input of any raw material in its own production. Sraffa writes for instance:

$$240 \text{ qr. wheat} + 12 \text{ t. iron} + 18 \text{ pigs} \rightarrow 450 \text{ qr. wheat}$$

Whereas mathematically this is equivalent to:

$$12 \text{ t. iron} + 18 \text{ pigs} \rightarrow 210 \text{ qr. wheat}$$

However, in the real world, in view of what economically (and mathematically) actually happens, we should rather write, for instance:

$$60 \text{ qr. wheat seeds} + 12 \text{ t. iron} + 18 \text{ pigs} \rightarrow 270 \text{ qr. wheat}$$

<footnote n="9">⁹ Sraffa's definition is the following: "the criterion is whether a commodity enters (no matter whether directly or indirectly) into the production of all commodities. Those that do that we shall call *basic*, and those that do not, *non-basic* products" (Sraffa, 1960, § 6, p. 8).</footnote>
Obviously, for the whole production system, there must be another separate equation (not represented here), for the production of wheat seeds, a commodity which is produced from wheat but is not the same as the wheat that has just been harvested, because it has to undergo some transformation in order to be transformed into seeds. We can note that there is no surplus of wheat, because all the wheat that is produced is used in the production of other goods, which can be final goods or other intermediate goods, like in the case of wheat seeds especially. There is no surplus of wheat seeds either, because all these seeds are employed in the production of wheat.

Sraffa's presentation recalled above thus prevents us from seeing the double origin of the apparent surplus in the sub-system of basic commodities, which is firstly the existence of non (humanely) produced goods which are not represented in the equations (here water, carbon dioxide, etc.) and secondly the use of part of these same basic commodities in another sub-system, i.e. the sub-system which corresponds to the production of non-basic commodities.

It remains nevertheless that globally, for the system of production as a whole, there cannot be any surplus. First because all of the final commodities, taken together, that appear on the right hand side of the sub-system of equations representing their production, are nothing more than the transformation of intermediate commodities, also produced, which appear on the left hand side of the same sub-system of equations. The apparent surplus of intermediate commodities which appear in the sub-systems of equations corresponding to their own production (either for the first type or for the second type) is fully absorbed in the production of final commodities. Therefore there cannot be any surplus of intermediate commodities.

As for these intermediate commodities, since they are not themselves produced ex nihilo, because production is a transformation, they are produced from the transformation of pre-existing elements which do not appear in the equations describing the system of production, because they are not the result of human production. To be sure nothing forbids to introduce these primary elements on the left-hand side of the equations, and therefore it would be theoretically possible to make them appear as such, if for instance we wanted to show the material balance of production. But then their value would have to be zero, because they have not been produced by man: no human labor has been spent in their production. This is the reason why they are generally not taken into account.

### 3. The existence of values and their domain of validity

What the whole second part has shown is that values are not only an intellectual concept, defined in the field of economic theory, but also an operational one, anchored in the real world, because it is perfectly possible in any circumstance, including the existence of joint products, to calculate values empirically. All that is needed to this end is to dispose of data external to economic theory regarding the techniques or methods of production, and among these techniques about the labor time, in physical units, spent when using these various techniques. Values are therefore not only a concept, but a reality.
Values were defined above in chapter 5 on the basis of Marx’s works, and through the concept of abstract labor, which we defined rather simply as labor reduced to its only dimension of time, which can be done in the context of the capitalist mode of production. Tis because wage labor is indeed generalized in the capitalist mode of production. However, in the real world this particular feature of labor which implies that it has to be measured as a function of time appears as soon as labor is paid in money, because this particular type of payment in most circumstances implies the specification of a monetary amount paid for a given quantity of time, which can be an hour, a week, a month or even a year.

But this feature is not limited per se to the capitalist mode of production. The monetary payment of labor against a wage is a phenomenon which took place in the real world in other modes of production, anterior to the capitalist mode of production: even in the Middle-Age wage labor already existed in cities, were the corporations of craftsmen used to enact rules for the payment of wages to companions: cathedrals builders did not work for free and were indeed paid not in kind but in the form of monetary wages. Workers received an agreed sum, paid in cash, at the end of the performance of their work. For instance on the building sites the organization of this work implied the measure of time spent to carry out the prescribed tasks: the first clocks appeared late on the sites, but the measurement of the time thanks to hour glasses has always been a concern of the employers.

What this means is that the concept of abstract labor can be generalized to a broader area than only the capitalist mode of production and therefore that the concept of value itself can be considered as still relevant for other types of modes of production, to the extent that wage labor exists in the corresponding societies. This presupposes the existence of money, because wage labor is paid in money, but money is also an institution which has existed for millennia and well before the development of the capitalist mode of production.

On this question we agree therefore with Frederick Engels, who in his supplement to Capital Volume 3, which he edited and published in 1894, wrote: "Starting with this determination of value by labor-time,...The most important and most incisive advance was the transition to metallic money... In a word: the Marxian law of value holds generally, as far as economic laws are valid at all, for the whole period of simple commodity production — that is, up to the time when the latter suffers a modification through the appearance of the capitalist form of production" (Engels, 1894).

This deserves to be emphasized, because using the concept of value can be useful for the theoretical analysis of the economic functioning of other modes of production. A good example of this type of analysis can be found in the works of Maurice Dobb, and in particular in his well-known "Studies in the development of capitalism", an important book that he published in 1946.

This also shows the tight link which exists between money and value, and which makes that one cannot exist without the other, in the real world as well as conceptually. This will become even more obvious in the next part, devoted to distribution and prices. Before examining this
question, a last point must however be touched upon, which is the independence of values
from the structure of the productive system.

4. Values are independent from the structure of the productive system

Let us first underline that by structure of the productive system we do not mean the technical structure of production, since - as we have shown so far - values are completely dependent for their calculation on the techniques which are used in the production process, and which are given as a set of data external to the economic theory as such.

What we mean by “structure of the productive system” is the structure of property over the entities, i.e. the enterprises, which are using and implementing these techniques in the course of the production process. For instance a supply chain for a particular type of commodities can often be fragmented among many enterprises: indeed a technical process can be cut in successive technical stages, each of them producing an intermediate commodity to be further transformed downstream in the process. Then each of these technical stages can generally be taken care of by a different enterprise. At the other end of the spectrum, all of these technical stages can be fully taken care of within a single enterprise, which is fully integrated for the production of this particular commodity. This does not necessarily mean that such an enterprise is necessarily in a situation of monopoly, because there can be several such integrated enterprises competing on the same market.

But this does not change the way of calculating values, because this more or less important
integration of the production process and the productive system has no bearing on the system
of equations giving the values of commodities, which as we have seen only reflects the
average production conditions for a particular commodity, whatever the number of enterprises
that are involved in its production and their location on the supply chain.

If this observation seems self-evident, it will take on its full meaning in part III which now
follows, devoted to distribution and prices.

References

Dobb, Maurice (1970). The Sraffa System and Critique of the Neo-Classical Theory of Distribution. *De Economist. Netherlands Economic Review*, vol. 118, Issue 4 (July 1970), pp. 347-362.

Engels Friedrich (1894). A supplement to Capital Volume Three. *Engels' Edition of the Third Volume of Capital and Marx's Original Manuscript*. New-York: International Publishers.
Marxist.org (1999) [online]. Available at:
https://www.marxists.org/archive/marx/works/1894-c3/supp.htm#law
Accessed 17 Aug 2018.

Sraffa, Piero (1960). *Production of commodities by means of commodities. Prelude to a critique of economic theory*. Cambridge (UK): Cambridge University Press.
# PART III - DISTRIBUTION AND PRICES

So far the analysis of commodities and of their values remained confined to the area of production, since it was possible indeed to demonstrate that value, as a property defining the equivalence of these commodities, which is nothing else than their common dimension, could be well defined within the theoretical boundaries of production. The moment has come however to go to the next stage, that of distribution, which can only physically take place once commodities have been produced. Any society or mode of production is indeed characterized by the particular form in which the product is distributed among its members. As we shall see, the first step toward this distribution occurs at the very stage of production, through the payment of wages. This is the case because in the real world wages are not evenly distributed to workers according only to the physical time during which they have been working. The influence of this phenomenon will be dealt with in a first chapter, chapter 10, devoted to the definition of social and monetary values. All the other chapters of this third part, from chapter 11 through chapter 16, will address the distribution of the product between workers and capitalists, and its influence on prices.
# Chapter 10. From physical values to social and monetary values

This chapter will allow us to go further in the definition of production as a social process, which involves various sorts of labor. This will lead to the discussion of the concepts of simple and complex labor, closely related to the question of the differentiation of wages. We will show that there cannot be any economic law of wage differentiation, which is a multi-factorial phenomenon. Finally this will allow us to explore the consequences of these findings on the theory of value and prices.

## 1. Production as a social process

So far we have focused and insisted on production as a technical and even a physical-chemical process. In this connection, labor has been considered essentially as an activity that can be quantified by using a physical variable, which is time, precisely because labor takes time and has therefore a time dimension. This is the reason why values cannot but have the dimension of time, and why we called them physical values. This is true whatever the society and historical time, as long as labor is accomplished in order to produce goods necessary for sustaining individual and social life, and to the extent that these goods are produced for exchange. Then labor itself can be considered as a commodity, because it is exchanged on a regular basis against a monetary wage, paid itself on a time basis.

But production is obviously much more than a physical and technical process: it is also a social process, socially organized in ways which change in accordance with the various modes of production. This explains that in the real world each country, society or culture has its own way to organize production socially and to value various kinds of labor differently. Moreover this organization and valuation usually change over time.

It is for instance possible to imagine, as a kind of thought experiment, two identical physical systems of production, with the same output of the same commodities, the same machines, the same workers, the same labor time spent for each production, and the same composition of the product, but with different monetary values for the commodities produced by these otherwise identical systems, because the differences in the social organization of production result in a different monetary valuation of different types of labor.

To go further in the understanding of production and value, it is therefore necessary to take into account these differences, and specify in more details the system under review. In the context of this work, we confine ourselves to the capitalist mode of production, where means of production are the property of individuals, and where labor is wage labor, with wages paid in money for given amounts of time. Moreover, although the amount of money wages and the period of time for which they are paid out are generally specified in contracts which exist before any work is carried out, and therefore before production starts, wages are generally paid once the various kinds of work corresponding to these contracts have been performed,
i.e. at the end of a specified working period, during which commodities have been produced.
It is also a reason why value and money are consubstantial.

These various features allowed the definition of abstract labor and the calculation of physical values, i.e. values expressed in physical time, whatever the differences in the nature of the different types of labor involved in the production of different commodities. This can be achieved without needing to know the precise amount of money paid for various kinds of labor. But as soon as a cycle of production is over and wages are paid at the end of this same cycle, then physical values are not enough to account for a thorough understanding of the system.

Indeed there is still one question which must be addressed, and this is the precise amount of money paid for the quantity of physical time spent in this production cycle. In the real world this amount is not purely proportional to the quantity of physical time, and therefore to the physical value of the corresponding product. This phenomenon results from the difference in wages paid for identical quantities of physical time, a discrepancy which in turn comes mainly from the existence of differences in the qualification of workers: it corresponds to the problematic that Marx called the reduction of complex labor to simple labor.

## 2. Simple labor and complex labor

To begin with, let us point out that for Marx simple labor does not mean unskilled labor, because any labor requires some kinds of basic skills, like for instance being able to read. Simple labor corresponds to this average, basic level of skills. Complex labor is therefore labor with additional skills, usually acquired by education and/or training, which allows to perform more complicated tasks than those performed by workers with this average basic skills.

Marx was well aware that this difference in the qualifications of labor had a direct repercussion on values. In Book I, Chapter 1, Section 2 (The Twofold Character of the Labor Embodied in Commodities) he wrote indeed: "A commodity may be the product of the most skilled labor, but its value, by equating it to the product of simple unskilled labor, represents a definite quantity of the latter labor alone¹⁵. The different proportions in which different sorts of labor are reduced to unskilled labor as their standard, are established by a social process that goes on behind the backs of the producers, and, consequently, appear to be fixed by custom. For simplicity's sake we shall henceforth account every kind of labor to be unskilled, simple labor; by this we do no more than save ourselves the trouble of making the reduction" (Marx, 1867, p. 32).

However and at the same time, Marx was excluding from the outset a potential way of making such a reduction, by making clear that he did not want to base his definition of values - in terms of complex labor reduced to simple labor, on the differentiation of wages: in note 15 of chapter 1 of Capital, appearing within the quote in the preceding paragraph, he considered indeed that he was speaking of "the value of the commodity in which that labor time is
materialized”, and that “wages is a category that, as yet, has no existence at the present stage of our investigation” (Marx, 1867, p. 54).

The consequence of this position is that there is not in Marx’s writings either a law or even a rule of transformation from complex labor to simple labor. This is well emphasized by J.-L. Cayatte in an article dating back to 1984, where he examines the question of simple labor and complex labor in Marx’s works. He finally summarizes Marx’s attitude as follows: “Between 1847 and the French version of the Capital, there is no doubt that Marx avoids the issue... However the result of it all is that a systematic comparison of the passages in which he tackles the difficulty leads to two different definitions of simple labor and five different definitions of complex labor. Should one be willing to establish a law of reduction of complex labor to simple labor, Marx's text should not be followed too closely and the inner logic of labor theory of value seems to be the one and only reliable guideline” (Cayatte, 1984, p. 220).

This is this inner logic which has been guiding us in our analysis. Indeed the whole approach followed so far in this book differs from this conception of Marx, since it has shown that wages and abstract labor, i.e. labor reduced to its dimension of physical time, and therefore wages and values, are consubstantial: labor value could not exist theoretically as a concept if wages did not exist simultaneously as the monetary payment of labor time. In opposition to Marx, this is the reason why we think that the differentiation of wages provides a proper mechanism, which is moreover a perfectly coherent one from a theoretical point of view, for the reduction of complex to simple labor.

If we try to understand why Marx rejected such an idea of starting conceptually from the actual differentiation of wages, it seems that it was for fear of being accused of empiricism, i.e. of developing a conception of knowledge based on empirical evidence, and because he thought that this kind of position would lead him to circular reasoning, since it would come down to starting from a reality that the theory is precisely aiming at explaining.

We think on the contrary that on this particular point Marx was wrong, not only because wages and values are conceptually linked, but also because the circular reasoning can occur only if we want to establish the laws of wages differentiation within the boundaries of economic theory, or more precisely of the theory of value itself. The circle is broken, though, if we acknowledge that wages differentiation is such a complex phenomenon that it can only be dealt with outside the realm of economic theory at such, by using a whole range of human sciences. Then, by becoming a set of variables that cannot be determined by economic theory alone, it becomes external to this theory, and can be considered as a given data.

### 3. The differentiation of wages as a first stage of distribution

This does not mean that the matter cannot be discussed. Understanding the difference between simple and complex labor remains in any case an important problem, because behind it the question raised is the difference in wages between workers with different levels of skills, which means that we are here at the heart of a problem which is a problem of distribution.
As we already indicated, distribution of the social product is usually considered as taking
place once production itself has taken place and once there are consequently commodities to
be distributed, in particular between workers and capitalists. But the differences which in the
real world characterize the level of wages in accordance with the qualification of labor lead us
to emphasize again what we already pointed out: that even before production takes place a
first distribution has already been organized through the existence of labor contracts
established in money amounts: it is a distribution between workers themselves, due to the
existence of a hierarchy in wages associated to different skills or positions within the
production process. This is the reason why this question was not touched upon in the second
part, but is dealt with only now in this third part of the book devoted to distribution and
prices, all the more so that wage is the monetary price of labor.

Since wages are paid in money, the payment of wages is the first theoretical step which occurs
in the transformation of physical values into monetary prices paid for commodities on the
market. At this stage however this first transformation only involves labor and corresponds to
the transformation of physical values measured in physical labor time into monetary values
corresponding to the payment of this physical labor time on the basis of predefined amounts
of money paid for predefined quantities of time. The payment of wages on the basis of a labor
contract takes the form of a payment of a certain number of monetary units per a predefined
unit of time, multiplied by a given quantity of these same time-units.

Wage has therefore a double dimension: as a money amount it is a pure dimensionless
number, i.e. a scalar, but since this amount cannot exist without the specification of a given
amount of physical time, always associated to this amount of money, it is by its very nature a
scalar per unit of time. Therefore, once all the payments for wages have been made, which
implies that wages are multiplied by quantities of time, physical values disappear to be
replaced by monetary values, i.e. the monetary value of the commodities which have been
produced with the intervention of these different kinds of labor. Since all types of labor are
not paid the same amount of wages per time unit, this necessarily introduces a distortion
between physical values and monetary values.

The question which remains to be asked, and if possible answered, is whether this distortion
can be theoretically understood and these monetary values theoretically determined. In other
words, we must ask ourselves whether there is a purely economic explanation to the
differentiation of wages. In Marxist language such a question aiming at understanding from a
theoretical point of view the range or hierarchy of wages can be considered as derived from
the question of the reduction of complex labor to simple labor.

4. An attempt by Marx to find a law of wage differentiation

After Marx expressly indicated, as we saw above in section 2 from a first quote from chapter
1 of “Capital”, that he wanted to “save (himself) from the trouble of making the reduction”
from complex to simple labor (Marx, 1867, p. 32), one would think that Marx would no more
touch upon this question. But he was not afraid of being confronted with contradictions,
because he decided nevertheless to go back to the problem, once it had been reformulated into
the question of the varying cost of labor-power, which is equivalent to the varying amount of wages, in chapter 6 of book 1, entitled "the Buying and Selling of Labor-Power".

Marx considers there that the key for understanding the reduction from complex to simple labor is education, and so he writes in Chapter 6 of Capital: "In order to modify the human organism, so that it may acquire skill and handiness in a given branch of industry, and become labor-power of a special kind, a special education or training is requisite, and this, on its part, costs an equivalent in commodities of a greater or less amount. This amount varies according to the more or less complicated character of the labor-power. The expenses of this education (excessively small in the case of ordinary labor-power), enter *pro tanto* into the total value spent in its production" (Marx, 1867, p. 121).

Marx thus considers that education and training costs constitute an element of the value of labor-power, of which wage is the monetary equivalent. In the following chapter he thus explains that labor-power which has benefited from such training has not only a greater value, but also creates proportionately greater values than unskilled labor-power. He writes: "All labor of a higher or more complicated character than average labor is expenditure of labor-power of a more costly kind, labor-power whose production has cost more time and labor, and which therefore has a higher value, than unskilled or simple labor-power. This power being higher-value, its consumption is labor of a higher class, labor that creates in equal times proportionally higher values than unskilled labor does"..."In every process of creating value, the reduction of skilled labor to average social labor, e.g., one day of skilled to six days of unskilled labor, is unavoidable. We therefore save ourselves a superfluous operation, and simplify our analysis, by the assumption, that the labor of the workman employed by the capitalist is unskilled average labor" (Marx, 1867, p. 137-138).

The solution thus proposed by Marx is not devoid of any logic, since it is based in a way on a kind of law of conservation of labor time or value. The labor carried out in order to obtain a qualification is indeed placed in reserve, and manifests itself as a source of value once it is used in a production process. If we remove innate talents from the causes of qualifications, because they appear mainly in the creation of non-reproducible goods like artistic performances or art works, for which the theory of value is irrelevant, there is no objection to assume that in the real world differences in qualifications come essentially from differences in prior education.

Following Ronald Meek in his "Studies in the labor theory of value" published in 1973, it is therefore possible to imagine an elementary calculation allowing the conversion of a quantity of complex labor into a quantity of simple labor. If *T* is the total productive period of a worker at the basic level of qualification, and if *t* is the duration of additional education and training needed to reach an upper level of qualification, then one hour of complex labor will be equivalent to $1 + rac{t}{T-t}$ hours of simple labor, if the productive period is the same. It might even be possible to refine the calculation, by considering that training time itself could be considered as a more and more complex labor as the training continues.
The problem with this approach is not that it is wrong *per se*, it is that it has a limited explaining capacity. Indeed if we try to apply the above formula in a developed country like France, there the minimum education and training time is approximately 10 years for a worker at the bottom of the salary scale, starting at the age of 16 a professional life of 44 years. These can be considered as representative conditions for simple labor. At the other end of the salary scale, an executive typically starts working around the age of 26 after a 20 year education and training time, leading to an MBA or a PhD. Even if we assume a shorter professional life of for instance 40 years for the latter, then the increased labor time content for an executive should be: $1 + \frac{10}{40-10} = 1,33$. This would justify a wage higher by 33% for an executive, whereas in France in 2013 monthly net salaries in the private sector ranged from 1200 euros for the first decile of workers to 8061 euros for the 99th centile, i.e. a spread of 572%. These data come from a study about wages in the private sector and public enterprises published in 2015 by the French National Institute of Statistics and Economic Studies (INSEE Première N° 1565, September 2015).

Similar demonstrations could be done in a number of other developed countries, which would show that salary scales are much wider and inequalities much higher among wage-earners than can be explained by simple differences in education. On top of the above finding it must be added that each country has its specific salary range: there are some countries like the Scandinavian ones with a narrower and more egalitarian range, and some countries like the US with a much wider range and a much more unequal distribution. It is thus clear from this demonstration that the range of wages has far more complicated determinants than the influence of education and training time. We will try in the next section to list a few of them.

# 5. Wage differentiation as a multi-factorial phenomenon

In the real world the question of wage differentiation is addressed by a number of institutions which play a role in implementing employment or unemployment compensation policies. By accumulating over the years huge amounts of data about wages, they have generally been able to construct some good models of wage predictability, linking the level of wages for any individual to a number of economic, social and institutional parameters. From that we can infer some coherent explanations about the main factors responsible for the hierarchy of wages at a given point in time and in a given country.

One such model has been developed in France by APEC (the French Association for the Employment of Executives). Its data are based on APEC's Survey of Professional Status and Compensation in 2014. The purpose of this survey is to collect information on the salaries of current managers. It was conducted with private sector executives. As many as 14 000 people responded on their professional status, training, career, position as private sector executive, and on the company in which they held this position. The salary questions covered both the fixed portion and the variable portion of compensation (variable salary, variable bonus, incentives and participation), as well as benefits and in-kind benefits. The results obtained enabled APEC to calculate and produce a set of data for annual gross salaries expressed in euros.
The model was developed using the method of variance analysis to determine the criteria that influence the salaries of executives and to measure this influence. These criteria are regrouped in three categories corresponding to the individual characteristics of a worker, his/her company, and his/her position. The wage estimation model takes into account all the coefficients of variation associated with these criteria and returns an average wage for each "profile". The model identifies 11 parameters, belonging to these categories, to explain the level of wages which can statistically be obtained by an average worker, and therefore his/her positioning on the salary scale.

The first of these three categories has thus to do with individual characteristics (work experience, education, etc.), and is therefore the training and profile of a worker: the nature (University or specialized graduate school) and level of his/her main degree, the number of years of his/her professional experience. A second category of parameters concerns the company where he/she works in: its sector of activity (for instance does it belong to the financial sector or the industrial one?), its size (number of its employees, importance of its turn over). A third category has to do with the position held by a worker within a company: the nature of the function (sales, marketing, accounting, etc....); the location of the company (in a big city or remote rural area); the nature of the employment contract (which can be more or less protective); the existence and importance of a budget to manage; the existence and importance of his/her hierarchical responsibility (particularly in terms of the number of workforce and managers that he/she supervises); whether the position is involved in international activities (with relations with subsidiaries, suppliers, customers abroad); and lastly whether the worker has or not a variable share in his/her salary.

What this empirical work shows unambiguously is that what Marx wrote in book I of "Capital" about education as the main determinant of wage differentiation, provides us with only a very limited view of the determinants of wages in the real world. Marx was thus wrong on this particular question. This survey gives a much more complex view of the determinants of wages. To be sure models such as the one that we just described can shed some light on the reasons for the differentiation that can be observed in salary scales in the real world. But there still remains factors that cannot be quantifiable, especially when we realize that salary scales that can be explained by this type of models differ from one country to another, for cultural, historical or political reasons.

Furthermore one important parameter is not taken into account by the APEC model, and it is gender. Indeed, in a country like France women's wages were 24% lower than that of men in 2014, according to the National Institute of Statistics and Economic Studies (INSEE). While inequalities decline slightly (the spread was 27% in 1995), they remain very marked among managers and high-income earners. Part of the wage gap is explained by the more frequent use of part-time and less-valued jobs: 44.8% of female jobs are concentrated in a few low-paying sectors such as public administration, health, education or social action, according to INSEE. Yet when comparing wages for equivalent conditions (sector, full-time, age, etc.), there remains a gap of 9.9% which constitutes "pure discrimination". These inequalities are all the more curious because in schools, girls perform better than boys. Among the younger generations, they are more educated: 31.3% of women aged 25 to 34 have a diploma
corresponding to a bachelor's degree and above, compared with only 26.4% of men. For the first time in 2013, 49% of managers entering the labor market are women. However, wage differences are noticeable upon graduation: for instance Sciences Po graduate females are paid 28% less than their male counterparts.

Such a gender pay gap is quite common in all countries, but it is stunning to note that even in the European Union, where some cultural homogeneity is supposed to exist among member countries, this gap varies significantly across EU Member States. For the economy as a whole, in 2015 women's gross hourly earnings were on average 16.3% below those of men in the European Union (EU-28) and 16.8% in the euro area (EA-19). Across Member States, the gender pay gap varied by 21 percentage points, ranging from 5.5% in Italy and Luxembourg to 26.9% in Estonia! These data come from the Gender pay gap statistics of EUROSTAT (March 2017). More generally these empirical data point out to the fact that the salary scale can be quite different from one country to another: to take but one last example it is for instance a well-known fact that engineers are better paid in France than in Germany, compared to ordinary workers.

The conclusion to be drawn from all these diverse sets of facts is that it is illusory to imagine that anybody may be able to construct a complete and coherent economic theory of wages differentiation. It is so because this phenomenon is the outcome of very complex social and political mechanisms. At a given point in time and in a given country, the differentiation of wages can only be explained by a multiplicity of factors, which have to do with ideology, institutions and the legal system – in particular the type of labor law, politics, culture, history. Obviously one such factor may also be the particular bargaining power of a given profession. The weight of trade unions can certainly play a role, but this may itself depend on the location of a particular group of workers within a supply chain, which may give a particular profession a capacity of nuisance by blocking the functioning of the economy as a whole.

It must thus be realized that two workers with exactly the same level of qualification, the same job and the same position in the hierarchy of two different companies may have two distinct levels of wage, due to a number of factors, like for instance gender or the location of each company. An important consequence of this recognition is that there is no unicity of the price of labor: exactly the same labor can have multiple prices. This is an empirical finding, which can also be explained theoretically, by statistical models such as the APEC model which was referred to, but should incite economists to modesty, and prevent them to think that an economic theory is able to explain everything which seem to belong to its field.

From all the above reflections we can now understand also why it remained impossible for Marx himself to establish a satisfactory definition (not to mention a measure) of the degree of complexity of labor. It is simply because like wage differentiation, to which it is closely related, labor complexity is such a complicated economic and social phenomenon that it cannot be explained by any universal law.

All this means finally that production at a theoretical level cannot be fully understood without realizing that it is underpinned by a first stage of distribution, i.e. the distribution of wages
among workers, which has multiple determinants. Owing to the ideological, cultural, political, institutional and historical nature of these determinants, which go well beyond the framework of economic theory, this phenomenon has to stay among the primary data of the theory, and as a largely exogenous data.

# 6. The consequences on the theory of value and prices

## 6.1. From physical to social values

In the theoretical description of the economic system we have to start from the fact that although production is a continuous process, it can be divided into cycles of a finite duration. Whatever such a cycle may be, it necessarily takes place, to begin with, on the basis of a particular set of techniques, which include the quantities and various types of labor involved in the production process. For production to take place producers need also to have hired workers to perform these different types of labor and entered into contracts establishing the precise money amounts of wages that have to be paid for these different types of labor. This presupposes the knowledge of the wage scale, but from the last section we can now fully understand that it is impossible to provide a purely economic theory of the differentiation of wages corresponding to this scale. This implies that the wage scale must therefore be taken as a given set of social data, imported from outside economic theory, as well as all of the technical data.

Establishing the level of wages in money terms implies that money is also necessarily present in the system, since it has to be used for the hiring of workers and the payment of wages. This also means that we must now go from the physical value of newly produced commodities to their monetary price, at which they are sold against money on the market. But there is an intermediate step, which comes from the fact that before posting the price of their commodities producers have first to account for the monetary amount of the wages that they have to pay, and which represents their monetized labor cost. Since these money payments are made in exchange for specified quantities of physical labor time corresponding ultimately to the physical values of commodities, it seems appropriate to call the corresponding amounts of money the monetary values of these commodities, so as to distinguish them from their monetary prices. Empirically and logically, these monetary prices will come into play one step further than these monetary values, if only because in the real world it takes some time after production before products are brought to the market to be sold. These monetary prices are thus conceptually derived from this monetary values, on which they need to be based.

The inevitable conclusion to be drawn from the recognition of these monetary values as factual elements of the real world is that the use of money wages is conversely the only method available for the conversion of complex to simple labor. Money wages provide indeed the weights that allow for the translation of monetary values to what should be called social values, i.e. physical values weighted by the various levels of wage corresponding to each different kind of labor. In order to keep the overall amount of labor time identical when going from physical values to these social values, the notion of simple labor should be replaced as we shall see now by that of average labor. The average labor time can indeed be defined
unambiguously as the labor time corresponding to the average wage per unit of time, i.e. the
ratio between the total wage bill and the total quantity of labor time expressed in physical
units.

Before showing it rigorously, and because it is instructive, let us start by briefly describing Morishima's attempt to address the joint problems of heterogeneity of labor and wage differentiation in Chapter 14 "The labor theory of value revisited" of his already mentioned book about Marx's Economics (Morishima, 1973, pp. 190-196), and by recalling the difficulties that he faces.

**6.2. Morishima and his difficulties with the heterogeneity of labor**

It is only at the end and in the very last pages of his book that Morishima indicates that: “we have so far assumed that all labor is homogeneous” (p. 190). He then recalls that Marx however recognized the heterogeneity of labor and decides to tackle the question himself. For that he does not try to criticize Marx (like we did earlier), but on the contrary wants to follow him closely by considering that skilled or complex labor is produced by education, which costs an equivalent in commodities. On this basis, Morishima intends to convert any kind of complex labor into simple, unskilled labor, and for doing so he uses a model where complex labor is itself produced by simple labor and commodities!

We will not expand on the model itself, but suffice it to say that Morishima discovers that the result of this model "is in conflict with his (Marx's) theory of exploitation, unless the conversion ratios are determined to be proportional to the wage rates of the different kinds of labor" (Morishima, 1973, p. 192). Indeed on the basis of our author's primary assumptions, and as he indicates: "we may have several groups of workers exploited at different rates, in contradiction to Marx's two-class view of the capitalist economy" (Morishima, 1973, p. 193). At first we could thus think that the conclusion that Morishima draws from this acknowledgment comforts our own position, since he continues as follows: "To avoid this difficulty, we have to abandon the scientific determination of the conversion rates by the formulas (21) and (22) and simply convert different kinds of labor into unskilled labor in proportion to their wages" (Ibidem).

Morishima then gives a formula which would allow this conversion of complex to simple
labor by using money wages, but it is to immediately reject its relevance: "In this case,
although the rate of exploitation is equalized throughout all kinds of labor, the values do not
satisfy the postulate of independence from market conditions and may easily fluctuate from
period to period as relative wages change".

It is thus on the basis of such an argument, combined with his prior criticism regarding joint
production (which was discussed previously), that Morishima then concludes his whole book
on Marx's Economics by rejecting the labor theory of value! But he is doubly wrong.

He is wrong first because labor is not produced. To be sure, labor can be considered as a
commodity because like other commodities it exchanges on the labor market against money.
But it is the only similarity it shares with other commodities. Indeed, and unlike other
commodities, it does not result from a transformation: labor is not a substance which would
itself be the end-product of the incorporation of other substances, thus transformed into labor!
Neither is there reciprocally any such thing as labor incorporated into commodities, because
the only scientific dimension of labor is physical time, and time is not a dimension that can be
incorporated in anything, which should remind us that a metaphor does not in itself constitute
a scientific statement. This means that what Morishima names in the above-mentioned quote
"the scientific determination of the conversion rates" (by the "production cost" of different
kinds of skilled labor), in spite of the fact that it is inspired by some writings from Marx, has
nothing scientific. So it is no surprise that this attempt was doomed to failure.

Morishima is also wrong on his second point, according to which wages are determined by market conditions, implying therefore, if money wages are used for the conversion of labor times of different kinds into standard (simple) labor, that values will depend on market conditions, whereas values are supposed ultimately to determine market prices. But as we already showed in the previous section of this chapter, although it is true that there is a labor market, in the real world and unfortunately for economists, the labor price on this particular market cannot be economically determined, i.e. determined only within the field and with the categories of economic theory.

Once this fact is recognized, it simply means that economic theory should not try to determine
some variables that are simply impossible to determine rigorously in its own field. Such a
lucid recognition has in itself nothing that should invalidate economic theory, as long as it
uses these thus external or exogenous variables as a basis for the determination of other
variables within its own theoretical field. The fact that, as Morishima writes it "values may
easily fluctuate from period to period as relative wages change" (p. 193) is not a cause for
abandoning a theory of value which produces such an outcome, but for analyzing the
consequences of these fluctuations, otherwise not economically explained.

To take only one example concerning France, the minimum wage was increased suddenly by
35 % and all the other wages by 7% on June 1st 1968, and 3% more on October 1st of the same
year. At the same time a reduction of the working time (number of hours worked in a standard
labor week) was initiated. This was not the result of any particular economic mechanism, but
of a general strike which triggered a tough negotiation ended by the "Grenelle Agreements".
This occurred in the context of a major political event: the May 1968 movement. Obviously
no economic theory will ever explain such wage fluctuations... On the contrary economic
theory has to integrate these fluctuations and their consequences in its theoretical architecture.

6.3. A solution based on money wages and average social labor

There is a solution to the integration of labor heterogeneity and wage differentiation (as well
as wage fluctuations) in the theory of value and prices. This solution requires to start from
wages as they exist in the real world, which must be taken as a given set of data, and to use
them as a basis for the definition of an average social labor. To show it, let us consider a
simplified system of values, inspired - up to a certain extent, by the system which we just
discussed (exposed in Morishima, 1973, pp. 191-193). Although we disagree with him, we
borrow some of his notations, which will make it easier to show where we differ from him. We assume that there are only intermediate commodities of the first type - with *h* such commodities, and *n* kinds of labor *l*<sub>*k*</sub> (*k* = 1,...,*n*). The total quantities of the different types of labor used in the system, measured in units of physical time, are represented by the column

$$
\text{vector } L = \begin{bmatrix} l_1 \\ \vdots \\ l_n \end{bmatrix}
$$

The total labor time spent in the system is thus: $L_{total} = \sum_{k=1}^{n} l_k = (l_1 + \cdots + l_n)$

Let us take $w_i$ as the money wage-rate for one equivalent unit of physical time corresponding
to these $n$ different kinds of labor (thus with also $i = 1, \dots, n$). The $n$ wage-rates are the elements
of the row vector:

$$
W = (w_1, \dots, w_n)
$$

The global amount of the wage bill spent in the whole system is therefore $W_{total} = WL$, from which we get the average wage-rate of the system $\bar{w}$, which is a weighted mean, the weights being the various quantities in physical time of the different kinds of labor, $l_1$ to $l_n$:

$$
\bar{w} = \frac{WL}{\sum l_k} = \frac{w_1 l_1 + \dots + w_n l_n}{l_1 + \dots + l_n} = \frac{WL}{L_{\text{total}}} \quad (1)
$$

From this equation it is clear that we can define one unit (for instance one hour, or one
month, or one year) of average social time, corresponding exactly to one identical unit (hour
or month or year) of physical time, because the average wage-rate $\bar{w}$ is precisely calculated
per unit of physical time (over the overall amount of this time). Knowing this average wage-
rate, we can now express all the various wage-rates in terms of this average wage-rate,
allowing us to define a new row vector, called $\bar{W}$, such as:

$$
\bar{W} = \left( \frac{w_1}{\bar{w}}, \dots, \frac{w_n}{\bar{w}} \right)
\qquad (2)
$$

Morishima does the same kind of conversion, but to convert *n* types of labor into what he
calls “labor *n* + 1 as the standard labor, i.e. the unskilled, simple labor or human labor in the
abstract” (Morishima, 1973, p. 193). Although his conversion is mathematically correct, it
makes it more difficult to define and carry out what we can call a transformation of physical
values into social values, determined on the basis of the average social labor associated to this
average wage-rate, since he confuses simple labor and abstract labor. In any case this is not
the crux of the matter, which is – as we have explained, what Morishima calls wrongly the
“market determination” of the various wage rates.
In fact $\bar{W}$ gives us the conversion ratios of all the various kinds of labor into an average social labor, because it is obvious from equation (1) above that we have:

$$L_{total} = \frac{WL}{\bar{w}} = \frac{w_1 l_1 + \dots + w_n l_n}{\bar{w}} = \frac{w_1 l_1}{\bar{w}} + \dots + \frac{w_n l_n}{\bar{w}} \quad (3)$$

$$L_{total} = \frac{w_1}{\bar{w}} l_1 + \dots + \frac{w_n}{\bar{w}} l_n = \bar{l}_1 + \dots + \bar{l}_n \quad (4)$$

To produce a unit of good j (with $j=1, \dots, h$), $a_{ij}$ units of good i and $l_{kj}$ units of direct labor k (with $k=1, \dots, n$) are required. Then the conditions of production can be represented by two matrices:

$$A = \begin{bmatrix} a_{11} & \dots & a_{1h} \\ \vdots & \ddots & \vdots \\ a_{h1} & \dots & a_{hh} \end{bmatrix} \quad \text{and} \quad R = \begin{bmatrix} l_{11} & \dots & l_{1h} \\ \vdots & \ddots & \vdots \\ l_{n1} & \dots & l_{nh} \end{bmatrix}$$

The quantities of direct labor of the different kinds necessary for the production of the h commodities can be converted in average social labor time, and then added so that we can define a row vector of these homogenized quantities of direct labor. Its h elements are:

$$\bar{l}_j = \frac{w_1}{\bar{w}} l_{1j} + \dots + \frac{w_j}{\bar{w}} l_{ij} + \dots + \frac{w_n}{\bar{w}} l_{nj} \quad (5)$$

It is easy to see that in the row vector below the $\bar{l}_j$ elements (with $j=1, \dots, h$), are obviously not the same as the $\bar{l}_i$ (with $i=1, \dots, n$), in equation (4) above, since the index j no more represents a particular kind of labor (converted into average social labor), but the global quantity of average social labor (homogenized labor), corresponding to various kinds of labor used directly in the production of a particular commodity j:

$$\bar{l} = (l_1, \dots, l_j, \dots, l_h) \quad (6)$$

Then we can also use the conversion vector $\bar{W}$ to calculate the amounts of average social labor time used directly and indirectly in the production of commodities:

$\bar{L} = \bar{W}R + \bar{l}$ where $\bar{W}$ is the conversion vector as defined above by equation (2), and where $\bar{W}R$ gives us the quantities of average social labor time used in the production of intermediate commodities i entering in the production of commodity j.

If $\bar{V} = (\bar{v}_1, \dots, \bar{v}_h)$ is the vector giving the social values of commodities, now expressed in average social labor time, they are then determined by:

$$\bar{V} = \bar{V}A + \bar{L} \quad (7)$$
We have thus demonstrated that the introduction of the heterogeneity of labor and of the associated wage differentiation does not prevent this system from determining values, which are still expressed in units of physical time but must now be called social values, because at the level of individual commodities they will necessarily differ from physical values as we had calculated them so far, i.e. on the only basis of physical labor time spent in their production, whatever its kind. Indeed these last ones did not take into account the heterogeneity of labor, i.e. the existence of complex labor of various kinds, in proportions which obviously differ from one commodity to another.

Moreover we can also very easily go from social values to money values, simply by multiplying social values expressed in average social labor time by the average wage rate!

In section 5 of chapter 6 we demonstrated the equivalence of total direct labor time and total product value. It follows that another important consequence of this transformation of physical values into social values and money values is that at the global level we can now identify the total physical value of the product to its total social value (which is nothing else than what we called $L_{total}$) and translate it into the total money value of the product, i.e. the overall wage bill, since by definition $W_{total} = L_{total} \bar{w}$!

This shows us that for the production process as such to be carried out, assuming obviously the existence of the capitalist mode of production, i.e. of workers and capitalists - these last ones being by definition the owners of existing fixed capital, all that is needed is the hiring of workers, whose labor alone (with the assistance of fixed capital), will be enough to produce intermediate commodities and final commodities: consumption goods as well as new fixed capital.

On the basis of the preceding demonstration we can now state two new principles, each of them with its own corollary:

**Principle 16:** In the real world labor is heterogeneous: there are many kinds of labor, which differ from one branch of production to another and according to the degree of qualification of labor. This is the reason why there is a differentiation of wages. But the precise range of this differentiation cannot be determined within the ambit of economic theory, since it depends on many factors external to this theory, and which are of a political, ideological, cultural and institutional nature.

**Corollary to principle 16:** the differentiation of wages, as it exists in the real world, must be considered as a given set of data for economic theory, which is one reason why this theory should rather be named political economy.

**Principle 17:** Taking as given the existing wage scale (which provides the level of wages for every kind of labor) allows for the conversion of different kinds of labor expressed in physical time into average social labor, or labor which is paid the average money wage, while keeping the same unit of physical time. This in turn makes it possible to calculate the average social values of all commodities, expressed in average social labor time.
**Corollary to principle 17:** Multiplying the social values of commodities by the average wage (in money units per unit of average social time) gives their money value at the stage of production.

**6.4. The nullity of monetary profits at the stage of production**

What we just highlighted, i.e. the correspondence between values as average social labor time and wages as the money payment of this labor time looks strongly like a kind of rehabilitation of Adam Smith's view of values as commanded labor. But everybody also knows that this theory has been rejected, and to some extent rightly so, because Smith did not clearly distinguish conceptually between the stage of production of commodities and the stage of their circulation (i.e. their exchange on the market), and therefore between their value and their price, as their exchange ratio against money on the market. It was thus easy to show that the monetary price of commodities included a profit on top of wages spent for the production of these commodities, a profit which moreover was not proportional to wages. It ensued that the price of a commodity could not be equal to the quantity of labor it commanded.

However, if we restore the distinction between a concept of value which is coherent at the sole stage of production and a concept of monetary price which is relevant at the stage of circulation, then the original idea of Smith regains its coherence, provided that we can show that the existence of profits as an income is itself not relevant at the stage of production, whatever the appearances may be.

In fact this is not so difficult to demonstrate, at macroeconomic level. First let us consider that capitalists start from scratch a new production process and that this production is carried out during a period corresponding to the normal interval for the payment of wages. For this process to be carried out they need money to hire workers, and have to borrow it from the banks. If the economy is fully integrated, at least for the production of each final commodity, with only one firm producing both such a final commodity and simultaneously all of the intermediate commodities needed to produce it, this firm will only have to borrow the amount of wages to be paid at the end of the period specified for such a payment. It is immediate that no profit will appear or be realized at the stage of production: it is only at the stage of circulation, when they have sold their final commodity on the market, that capitalists will be able to realize a monetary profit, by selling their product at a price higher than the wages paid for its production. From these proceeds they will then be able to repay their debt to the banks.

If now the economy is disaggregated, firms producing final commodities will have to buy intermediate commodities that they need for this purpose from other firms producing these commodities. And they will have to buy them at a monetary price including a monetary profit. Thus, on top of borrowing to pay for their own wages, these first firms will have to borrow in order to pay for the purchase of these intermediate goods, at a price including not only the amount corresponding to the wages paid for the production of these intermediate goods, but also the amount corresponding to a profit for their producers, and will thus have a corresponding and additional debt of an equal amount.
The share of the proceeds from the debt of these first firms utilized to pay for the sales of the second ones, by definition (since they are spent) cannot be used for the repayment of the corresponding debt and therefore cannot cancel globally. The phenomenon is not the same at the next level, because the second tier of firms that are paid for selling intermediate goods to final producers do not need to borrow for paying their workers' wages and their own suppliers: this payment is included in the price that they have charged to the first firms, equal to the amount that they have received. But the amount that on turn they have to pay to their own suppliers offsets the amount received by these suppliers, and so on and so forth at each stage of the production process: to each realization of sales we necessarily have a corresponding payment to the next level of suppliers, which therefore reduces the global profits by an identical amount. What remains nevertheless at each production level is a net monetary profit.

Ultimately, when all these exchanges have taken place, the amount of net monetary profits realized by all the suppliers of intermediate goods is necessarily equivalent to and cancelled by the corresponding amount of debt that has to be repaid by the initiators of the process: the producers of final commodities, because precisely they had to borrow for paying an amount which included all of these net profits. As long as they have not sold their commodities as final goods in the sphere of circulation, this debt is exactly offsetting the sum of net profits realized by the suppliers of intermediate goods, for an identical amount. And it is only outside of the sphere of production, with the sale of final commodities on the market, that the repayment of their debt will be made possible and will allow for the disappearance of these negative profits.

In fact one can summarize this whole process by saying that the monetary profits realized by the producers of intermediate commodities are realized at the expense and correspond to negative profits of the producers of final commodities, who are their only buyers. Although it may seem somewhat as a caricature it is a logical and mathematical truth.

It follows from this demonstration that no net profits are created and can exist at global or macroeconomic level at the stage of production: taken globally, capitalists as a whole cannot realize any profits at this stage, through the exchange of intermediate commodities between themselves. This means that as an income, profits cannot be a category of production, and that the money value of the product is made only of wages.

The situation is quite different as regards wages: unlike profits, wages have a purchasing power at the very moment when they are paid to workers, and this for two reasons: first because no worker would accept to work if the wage that he is going to be paid by contract had no purchasing power, and second, because workers do not have to borrow the money that they receive as wages for the payment of their labor, and thus have no corresponding debt to be repaid once they receive the money amount of their wage. Since the main characteristic of an income is to have a purchasing power, wages are therefore an income at the stage of production, and they are the only income, which is completely coherent with the result achieved above at the end of sub-section 6.3., and according to which at the stage of production the global wage bill is equivalent to the money value of the product.
6.5. A solution which is reminiscent of Keynes

The reader who is familiar with the “General Theory” and remembers its chapter 4 “the Choice of Units” will note that the point to which we have arrived is not so far from the position adopted by Keynes, who writes: “In dealing with the theory of employment I propose, therefore, to make use of only two fundamental units of quantity, namely, quantities of money-value and quantities of employment. The first of these is strictly homogeneous, and the second can be made so. For, in so far as different grades and kinds of labor and salaried assistance enjoy a more or less fixed relative remuneration, the quantity of employment can be sufficiently defined for our purpose by taking an hour's employment of ordinary labor as our unit and weighting an hour's employment of special labor in proportion to its remuneration; i.e. an hour of special labor remunerated at double ordinary rates will count as two units. We shall call the unit in which the quantity of employment is measured the labor-unit; and the money-wage of a labor-unit we shall call the wage-unit [5]. Thus, if *E* is the wages (and salaries) bill, *W* the wage-unit, and *N* the quantity of employment, *E* = *N* × *W* (Keynes, 1936, p. 41).

One needs to realize that to become homogeneous and thus assume the status of true
measurement units these quantities of employment must necessarily be expressed in time
units (of physical time), and therefore be measured in weeks, months, or years of labor time.
Then there is an obvious connection between this particular choice of unit and the labor
theory of value, although Keynes was careful not to use this expression, which would have
made him appear as a disciple of Marx, and be even more misunderstood! To be sure, Keynes
does not come to this conclusion on the basis of the same approach as that developed so far in
this work. Indeed, against what he calls “the postulates of the classical economics”, he wants
to follow a more pragmatic and empirical approach, and develop a theory which accounts for
the real world. However it seems very clear from this quote that on the basis of somewhat
different assumptions he arrives at a similar result.

What deserves also to be noted is that to address the question of the heterogeneity of various
"grades and kinds of labor" Keynes does not enter into theoretical considerations, but simply
weights the different kinds of labor "in proportion to their remuneration", again almost
exactly like we did in subsection 6.3. above. A first difference lies in the fact that we provided
all due justification for doing so. A second difference comes from the fact that Keynes uses
what he calls ordinary labor, equivalent to Marx's "simple labor", as his unit of labor time -
and this is an empirical notion, whereas we use average labor, i.e. labor paid the average
wage, which is an abstract concept, as our standard of labor time.

As regards the second “unit of quantity”, as Keynes calls it, this is for him the quantity of
“money-value”, and the use of this expression shows in passing that there is some confusion
in Keynes’s mind about the distinction between the terms “value” and “price”. Indeed Keynes
seems most of the time to consider them as equivalent. This might be due to the fact that the
General Theory is not a theory of production as such, but a monetary theory of production
(Keynes, 1933). For him there is no production without money, because wages are paid in
money, and obviously this money is not a commodity and even less a share of the product:
Keynes's theory is fully incompatible with Sraffa's theory. As soon as wages are paid in money, a quantity of money can be multiplied by the quantity of employment, and everything can be measured in money. Prices are thus monetary prices, and the concept of value as distinct from the concept of price does not seem to be needed by Keynes: it does not come to his mind that its quantity of employment could define the value of the product, in terms of physical time, independently of its monetary value, measured by what he names the wage bill, which differs necessarily from the monetary price of this product, because of the existence of profit.

This absence of a clear distinction between monetary value and monetary price certainly explains the very complicated developments devoted in the General Theory to the supply price of the output, at macroeconomic (aggregate) level, and to the user cost, which requires a whole appendix to chapter 6 on "the Definition of Income, Saving and Investment". In fact Keynes introduces this question in chapter 3, where it is clear that for him the only factor of production is labor, with the aggregate supply function defined only as a function of employment: $Z = \phi(N)$. Then what he names "supply price" corresponds to what we have called the monetary value of the product: we have shown that it is nothing else than the overall wage bill. As for the monetary price of the product, as distinct from its monetary value, it corresponds to what Keynes calls *D*, i.e. "the proceeds which entrepreneurs expect to receive from the employment of *N* men, the relationship between *D* and *N* being written *D* = *f*(*N*), which can be called the *aggregate demand function*" (Keynes, 1936, p. 25).

It must also be understood that both terms used by Keynes, i.e. supply and demand, correspond to the supply and demand of *final* commodities only, to the exclusion of any intermediate commodities. It means that for the producers of final goods (be they consumption goods or capital goods), if they are not integrated enterprises, the "supply price" in terms of employment does not correspond only to their own employment, but also to the employment of the producers of the intermediate goods that these producers of final goods have used in the production of these final goods.

This is something that Keynes had understood, even though his explanations are somewhat confused, as it appears from footnote 3 to this same chapter 3: "The reader will observe that I am deducting the user cost both from the proceeds and from the aggregate supply price of a given volume of output, so that both these terms are to be interpreted net of user cost; whereas the aggregate sums paid by the purchasers are, of course, gross of user cost. The reasons why this is convenient will be given in Chapter 6. The essential point is that the aggregate proceeds and aggregate supply price net of user cost can be defined uniquely and unambiguously; whereas, since user cost is obviously dependent both on the degree of integration of industry and on the extent to which entrepreneurs buy from one another, there can be no definition of the aggregate sums paid by purchasers, inclusive of user cost, which is independent of these factors. There is a similar difficulty even in defining supply price in the ordinary sense for an individual producer; and in the case of the aggregate supply price of output as a whole serious difficulties of duplication are involved, which have not always been faced. If the term is to be interpreted gross of user cost, they can only be overcome by making special assumptions
relating to the integration of entrepreneurs in groups according as they produce consumption-
goods or capital-goods which are obscure and complicated in themselves and do not
correspond to the facts. If, however, aggregate supply price is defined as above net of user
cost, the difficulties do not arise. The reader is advised, however, to await the fuller discussion
in Chapter 6 and its appendix" (Keynes, 1936, p. 33).

To better understand what Keynes intends to say, even though he does not express it very
clearly, let us make a thought experiment, and imagine a capitalist economy that is fully
integrated, with therefore only one firm producing everything, i.e. both consumption goods
and capital goods, as well as all of the intermediate goods needed to produce these final
goods. Then we cannot but realize that the factor cost is only made of wages, which constitute
what Keynes calls the aggregate supply price and what we call the monetary value of the
product, i.e. the wage bill. Moreover it is obvious that in a capitalist mode of production by
definition workers do not buy capital goods. If we add the assumption that capitalists do not
buy consumption goods, knowing that capital goods will not have to be bought on the market
by the integrated firm (since it has produced them and already owns them), then the only
goods sold on the market are consumption goods.

The situation is such that, if all these goods are sold, and the market is cleared, the overall
price of these consumption goods must necessarily be equivalent to the wage bill W, which
constitutes for Keynes “the proceeds which entrepreneurs expect to receive from the
employment of *N* men” or “the aggregate demand” D, which is thus equal to this supply price.
However the factor cost of these consumption goods alone does not correspond to the
employment of *N* workers, but only to the employment of workers for the production of
consumption goods. If we name it *Nc*, and the corresponding wage-bill for these workers *Wc*,
we can see that these proceeds include a notional profit, which amounts to $Wp = W - Wc$.

It results from this demonstration that the disaggregation of the production system which is
the norm in a capitalist economy hides the fact, that Keynes had recognized at least implicitly,
that profits are a transfer income. We can see also why for Keynes fixed capital does not
transfer any value to the product, which derives from the deduction of the user cost from the
price of output, being understood that the price of output includes nevertheless all the new
fixed capital produced in each period<fnref n="10" />, because this fixed capital consists of final goods. All
this will be made even clearer in the next chapter, devoted to the realization of the product.

We have now arrived at a point where we can express another principle:

**Principle 18: The wage bill is the measurement of the monetary value of the product.**

From what we have discovered so far we can now understand the fundamental importance of
wages as an economic variable: it is indeed the determination of wages and of their
differentiation, by a number of factors which are mainly out of the reach of economic theory,
which allows this theory to determine simultaneously the monetary values and the social
values of commodities and of the product.

<footnote n="10">¹⁰ It also includes the consumption of intermediate goods, which contributes to obscuring the problem.</footnote>
References

APEC (2015). *Evolution de la rémunération des cadres en 2014* (Evolution of executive compensation in 2014). Paris: *Association pour l'emploi des cadres* (Association for the employment of executives).

Cayatte, Jean-Louis (1984). *Travail simple et travail complexe chez Marx* (Simple labor and complex labor in Marx's works). *Revue économique*, volume 35, N°2.

EUROSTAT (2017). Gender pay gap statistics. Data from March 2017.
https://ec.europa.eu/eurostat/statistics-explained/index.php/Gender pay gap statistics
Accessed 5 Sep 2018

INSEE (National Institute of Statistics and Economic Studies) (2015). *Salaires dans le secteur privé et les entreprises publiques* (Wages in private sector and public enterprises), INSEE Première N° 1565, September 2015. Paris: INSEE.

Keynes, John Maynard (1936). *The General Theory of Employment, Interest, and Money*. Cambridge (UK): Cambridge University Press.

Keynes, John Maynard (1933). *A Monetary Theory of Production*. In D. Moggridge (Ed.), *Collected Writings of John Maynard Keynes* (1973), vol. XIII - The General Theory and After, Part I - Presentation (pp. 408-411). London: Macmillan.

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 17 Aug 2018

Meek, Ronald Lindley (1956). *Studies in the labor theory of value*. 2nd Edition 1973. London: Lawrence & Wishart.

Morishima, Mishio (1973). *Marx's Economics: a Dual Theory of Value and Growth*. Cambridge (UK), London: Cambridge University Press.
# Chapter 11. Distribution, surplus-value and profit

In the previous chapter we saw that the first step of distribution, i.e. distribution between
workers themselves, takes place at the stage of production, and in fact even before production
starts, when wages are negotiated and their level established. We also explained why this
particular distribution depends on a number of factors which do not pertain to economic
theory as such. We intend now to engage in an attempt to understand a new phase of
distribution, i.e. distribution between workers and capitalists. We will show that it is a
distribution of commodities, but also of surplus-value, a concept which will be explained
through a particular mechanism, i.e. the transformation of values into monetary prices. This
process itself is at the origin of the creation of monetary profits, which will be demonstrated
through a simple model, based on simple reproduction.

## 1. Distribution as distribution of commodities and surplus value

Whatever the mode of production, and in any society, distribution is not only the distribution
of monetary income, but ultimately the distribution of commodities which have been
previously produced and (in a capitalist economic system) are thus available to be bought on
the market, between the members of this society. What Marx highlighted is that it is the
distribution of two quite different categories of commodities: consumption goods, on the one
hand, and means of production or capital goods, on the other hand. Due to the specific nature
of means of production, and to their appropriation by those who are by definition named
capitalists, which allows them to produce goods to be sold on the market and to hire workers,
distribution is also the distribution of men into social classes exercising distinct functions in
the production process. This point has been emphasized by Althusser in his book written with
Balibar "Reading Capital", published in English in 1970 (Althusser, 1970, p. 167).

Because of these different social functions, it is an obvious empirical fact that workers cannot
collectively obtain on the market the share of the product made of means of production, or
fixed capital (if they did so, they would by definition no longer be workers, but would have
become capitalists), and therefore that they only buy consumption goods. In fact in the real
world they only buy a part of these goods, because capitalists, as normal human beings, also
need to buy consumption goods). This is the case because the wages that they earn never
allow them to buy back the whole quantity of commodities that they have produced, but only
a share of these consumption goods sold on the market by capitalists.

Such a result is achieved through a very simple mechanism: the monetary price of the
consumption goods obtained by workers, which is equivalent to the total wage bill (assuming
at this stage that it is wholly spent) is greater than the monetary value of these same
consumption goods as defined in the preceding chapter, i.e. the total sum of direct and indirect
wages needed to produce these consumption goods. In monetary terms this difference is called profit. If we stay at this theoretical level, i.e. at the level of circulation of money and
commodities, we will remain blocked in the classical economic world of Adam Smith and
Ricardo.

But the novelty of Marx is to go back to the world of production and values, which allows him to realize that the social value of these consumption goods obtained by workers, in terms of labor time, is lower than the total social value of all the goods that they have produced: this difference, between the overall social value of the product and the social value of the consumption goods obtained by the workers, is nothing less than what Marx names “surplus-value”. In his book “For Marx”, a collection of previously published articles, published in English in 1969, Althusser considered this theoretical finding as part of an “epistemological breakthrough”, not only with classical economists, but “in the history of Marx’s thought, a basic difference between the ideological ‘problematic’ of the Early Works and the scientific ‘problematic’ of Capital” (Althusser 1969, p. 13). As this same author writes also in his previously cited book “Reading Capital”: “the sensitive point in this revolution concerns precisely surplus-value. Their failure to think it in a word which was the concept of its object kept the classical economists in the dark, imprisoning them in words which were merely the ideological or empirical concepts of economic practice” (Althusser, 1970, p. 148).

If money prices were equal to money values, then workers would be able to buy back the totality of the product, i.e. both consumption goods and fixed capital goods: the wage bill would be equivalent to the money price of the whole product. The condition for the theoretical existence of surplus value is therefore the distinction between values and prices, and more precisely values and money prices. It is because they were never able to make this clear distinction that Smith and Ricardo failed to come to the same conclusion as Marx’s. To be sure this difference is easy to understand once we realize that values and prices are not defined at the same level of the economic system: values are defined in the sphere of production, whereas prices are relevant in the sphere of circulation, where distribution takes place. This explains why they do not pertain to the same measurement system, and have distinct dimensions: time for values and no dimension for prices, which are pure scalars. But this difference still exists between money values (i.e. labor-time converted into money through the payment of wages) and money prices: the money value of the product differs from its money price, precisely because of the existence of profits.

Morishima is therefore perfectly right when he says in chapter 5 “Surplus-value and exploitation” of his previously cited book that: “in the capitalist economy (unlike the society with simple commodity production) values and prices, in general, no longer coincide; they should be distinct. For this reason Marxian economics, unlike orthodox economics, has dual accounting systems: one system in terms of value and the other in terms of price. But many people from both camps (such as Sweezy, Joan Robinson and Samuelson) have confused the two, since Marx himself sometimes confused them.” (Morishima, 1973, p.46). Morishima provides indeed a striking example of this confusion with this citation of Marx, who writes in Volume 3 of Capital: “if this surplus value is related to the total capital instead of the variable capital, it is called profit, *p*, and the ratio of the surplus value *s* to the total capital *C*, or *s/C* is called the rate of profit, *p*’. Now substituting for *s* its equivalent *s’v*, we find *p’* = *s’v/C* =
$s'/c+v$, which equation may be expressed by the proportion $p': s' = v/C$; the rate of profit is related to the rate of surplus-value as the variable capital is to the total capital" (Marx, 1894, chapter 3, § 3).

Marx's mistake comes from the fact that surplus-value is clearly defined in terms of values and not of prices, and that therefore the rate of surplus values $s/v$ belongs to the domain of values, as well as the rate $s/C$, because $C$, like $s$, has to be expressed in value terms (it is the value of total capital) and not in terms of prices: if $C$ were the price - and not the value, of constant capital it would itself incorporate a profit, and the ratio would be a ratio between quantities expressed in different dimensions: labor time for $s$ and scalars (prices) for $C$.

To keep Marx's terminology, this explains why the ratio $s/C_v$ (the ratio of surplus value to the value of total capital in value terms, which we thus write $C_v$) is necessarily different from what is usually called the rate of profit in economic theory, i.e. the ratio of monetary profit over the monetary price of capital $p/C_p$. In the rest of this book, we will stick to the usual sense of the concept of rate of profit, called $r$, as a ratio of quantities expressed in money. And since the concept corresponding to the ratio $s/C_v$ expressed in value terms does not seem to exist, but is nevertheless closely related to the previous concept, we will call it "rate of profit in value" and call it $r_v$.

These remarks inevitably raise the problem of the transformation of values into prices, which has provoked an endless debate among economists.

## 2. From values to prices, or the transformation problem

We will not expand too much on this question, which Marx addresses in chapter IX of book 3 of Capital, entitled "Formation of a general rate of profit (average rate of profit) and transformation of the values of commodities into prices of production". In trying to resolve it, he made a number of mistakes.

Indeed he starts from the postulate that there is an identity between the total sum of profits and the total surplus value and assimilates the production cost (in terms of prices) to the sum of variable capital (or value of wages) plus the value of constant capital (the value of intermediate goods plus the value transferred by fixed capital). Then it is clear that he can only get the sum of values equal to the sum of prices.

This postulate implies that Marx's solution comes down to allocate the same overall surplus-value differently in the price system as it is in the value system: instead of being proportional to the value of labor-power (what he calls variable capital, or the value of wages), surplus-value becomes proportional in the price system to the value of the total capital (variable plus constant capital). We know that all this cannot but be erroneous, if only because we are in two systems of different dimensions, social (or physical) time for values and scalar for prices, and that these two systems are therefore wrongly mixed.
In any case Marx himself was aware that his method was wrong, or to say the least approximate, when he wrote in chapter IX of Book 3 of “Capital”: “This statement (that the sum of the prices of production of all commodities produced in society - the totality of all branches of production - is equal to the sum of their values, author’s note) seems to conflict with the fact that under capitalist production the elements of productive capital are, as a rule, bought on the market, and that for this reason their prices include profit which has already been realized, hence, include the price of production of the respective branch of industry together with the profit contained in it, so that the profit of one branch of industry goes into the cost-price of another” (Marx, 1894, p. 122).

The first economist to propose a solution to this problem was Bortkiewicz, a Russian economist, in an article published in 1907: “On the Correction of Marx's Fundamental Theoretical Construction in the Third Volume of *Capital*”. This article was translated by Sweezy and published in English as an appendix to Böhm-Bawerk's book. In this article Bortkiewicz rightly indicated that “This solution of the problem cannot be accepted because it excludes the constant and variable capitals from the transformation process, whereas the principle of the equal profit rate, when it takes the place of the law of value in Marx's sense, must involve these elements” (Bortkiewicz, 1907, p. 201).

We will not develop in details Bortkiewicz’s solution. Suffice it to say that he exposes a system with three branches or sections (the same as Marx in book 2 of Capital), producing respectively the means of production (constant capital), wage-goods (variable capital) and luxury goods. He chooses the price of luxury goods as the unit of measurement. The system is in a state of simple reproduction, which makes it in fact quite similar to Sraffa’s system of production prices.

This system allows Bortkiewicz to calculate the rate of profit and the prices, and to show that it does not give Marx’s result according to which the sum of values is equal to the sum of prices. Obviously he finds like Sraffa that the rate of profit is determined only in the first two sections, that produce constant capital and variable capital, which is contradictory to Marx’s formula, according to which it is the totality of constant capital (including capital in the luxury goods section) which plays a role in the determination of the rate of profit. It is therefore no surprise that most of Bortkiewicz’s article is devoted to the criticism of Marx’s formula to calculate the profit rate: $r = \frac{s}{C} = \frac{s}{c+v}$.

The ultimate avatar and logical outcome of Bortkiewicz’s approach is Sraffa’s system of production prices, in which production prices and the rate of profit can be determined directly on the basis of technical data, in terms of physical quantities, including the quantities of wage-goods consumed by workers. And this can be done even before the calculation of values.

This conception, which has been designated as the inverse transformation theory, was expressed in the most complete manner by Samuelson, in an often cited article: “Understanding the Marxian Notion of Exploitation: A Summary of the So-Called Transformation Problem Between Marxian Values and Competitive Prices”. In this article,
published in 1971, he wrote: "just as Marx turned Hegel on his head, we turn Marx upside down, and can say in a dozen repetitive ways that this total of profit is not allocated by the value system according to where it was 'really produced' but rather falls to the share of each aliquot part of the total social [variable] capital out of the total social profit" (Samuelson, 1971, p. 417). And also, one page further: "a theory of exploitation wage can be based solely upon the analysis of profits and prices (with all the Leontief-Sraffa algebraic complications that are involved in these real-world relations)" (Samuelson, 1971, p. 418).

Since we have criticized at length the system of production prices in chapters 3 and 4 of this book, we will not repeat this criticism here. Let us just state again that in any case values, on the one hand, and production prices, on the other hand, cannot be compared, because they do not have the same dimension: values, determined at the stage of production, have the dimension of physical time, and production prices have the dimension of the standard commodity, i.e. an heterogeneous and permanently changing bunch of commodities.

Values and production prices belong indeed to two completely different universes. Moreover the system of production prices suffers anyway from a number of internal flaws that we have fully exposed. Our criticism is quite sufficient to sweep away all this problematic of inverse transformation, and all the more so that the theory of production prices is of no help to understand how profit is formed and realized, a question that we will now address.

### 3. The realization of profits, in a simplified model

To correctly address the question of the formation and realization of profits, it is better to adopt a step by step approach. Since the realization of profits implies the sale of commodities against money, which is the first step of the reproduction of the system, we will start with a simple model of reproduction, which we will first define in the next subsection.

#### 3.1. A model based on simple reproduction

This model is based on the following assumptions, which correspond to the mode of exposure of Marx:

- The system of production is the pure capitalist mode of production, in the sense that there are only workers and capitalists.

- Capitalists are the owners of the means of production, and they employ salaried workers. As far as production is concerned, they are represented by firms, corresponding to what national accounts call corporations or quasi-corporations, non-financial as well as financial, and household firms, provided that the latter employ at least one salaried worker. But the capitalists are also individuals who are the ultimate owners of the firms in question, and as such they are part of households in the sense of national accounts.

- Workers by definition do not own any means of production, and sell their labor power to the capitalists, against payment of a money wage: they are not paid in consumer goods. This point is important, and in particular introduces an important difference between the
capitalist mode of production and other modes like the slave mode of production or the
feudal one. The question of the nature of money has already been discussed, and we just
need to point out once more that money is by no means a commodity.

- The economy consists of two sections, section I, which produces fixed capital goods, and section II, which produces consumer goods. The question of whether there are different production methods is not considered relevant, because the choice of techniques has already been made and the system self-reproduces itself from period to period. We are in the theoretical world of commodities and abstract labor and as a result, physical goods, which Marx and others call use-values, do not appear as such in the model. We must emphasize the fact that these two sections have nothing in common with the sectors that appear in the theory of production prices: the sector of basic goods (of whatever type) and the sectors of luxury goods. Indeed both fixed capital and consumer goods are luxury goods in the sense of the theory of production prices, which means that, once they have been produced, they never re-enter the production process to be transformed through it.

- Surplus-value is the difference between the total value of the whole social product and the value of the consumer goods which workers can obtain by exchanging their money-wages for commodities, called conventionally the value of their labor-power. The rate of surplus-value is thus the ratio between the surplus-value at global level and the value of labor-power.

- The rate of surplus-value as defined above, and sometimes called the exploitation rate, is uniform in the two sections: surplus value is indeed considered as a global social phenomenon, which concerns all the workers taken collectively, and not as an individual phenomenon. This rate is, however, uniform only within the framework of a given social system, corresponding to a separate and autonomous political organization and therefore usually to a State.

- It is assumed for the moment that capitalists do not consume and that workers do not save, which simplifies the reasoning, but as will be seen is not without consequence on the demonstration. This assumption will then be abandoned when the consumption of capitalists is introduced into a second, more complex model.

- All profits are saved and invested in the section where they are formed, which implies that there is no transfer of savings between the two sections.

- On the other hand, the assumption according to which workers do not save will be preserved, for it is consubstantial to the very definition of workers as salaried workers employed by capitalists who own the means of production. Since saving is by definition identical to income less consumption, it follows that if workers saved, this saving would by the same token be used to purchase goods other than consumer goods and hence for buying means of production. But in this case workers would ipso facto turn into capitalists, which would be contradictory!
- This raises the question of the status of real estate, whose value is considered in the national accounts as the "gross fixed capital formation of households", which is supposed to legitimize the treatment of their counterpart in income as constituting "households savings". In fact, dwellings are durable goods, and even if they can obviously be leased, they cannot be considered as capital in the sense of Marx: capital is in fact such because it is not only a means of production, but the support of the employment of salaried workers who make it produce (which is obviously not the case of housing, even when rented). Real estate status, and consequently that of household savings, should therefore be modified in national accounts. This question of the conceptual status of real estate is discussed more comprehensively in an article that I published on Piketty's best-selling book (Capital in the 21st Century): "Again on Piketty's *Capital in the Twenty-First Century* or Why National Accounts and Three Simple Laws Should not Be a Substitute for Economic Theory" (Flamant, 2015a, pp. 24-25).

- Moreover, means of production correspond only to fixed capital. This implies that there is no section specifically producing circulating capital, which is produced within each of the two sections I and II. In each section production is thus integrated, in the sense that each section produces, with the aid of means of production and labor, the totality of its circulating capital, i.e. intermediate goods necessary for the production of its final goods, and that disappear as such (in their original form) in the course of the production process (upstream of it), in order to reappear downstream, after transformation, by being incorporated into these final goods. In his book published in 1981: "Structural change and economic growth: a theoretical essay on the dynamics of the wealth of nations" Pasinetti indicates that: "in a vertically integrated model, the criterion is the process of production of a final commodity, and the problem is to build conceptually behind each final commodity a vertically integrated sector which, by passing through all the intermediate commodities, goes right back to the original inputs" (Pasinetti, 1981, p. 113). He shows that for every final good one can obtain the vector of the vertically integrated circulating capital stock of the corresponding branch, which can be regarded as a kind of composite good, called vertically integrated production capacity unit. The same procedure applies to labor coefficients, so that for each final good it is possible to obtain the integrated labor quantity and the corresponding unit of vertically integrated production capacity. On this point Pasinetti was relying on Sraffa, who in appendix A on sub-systems of "Production of commodities..." indicates: "Although only a fraction of the labor of a sub-system is employed in the industry which directly produces the commodity forming the net product, yet, since all other industries merely provide replacements for the means of production used up, the whole of the labor employed can be regarded as directly or indirectly going to produce that commodity" (Sraffa, 1960, p. 105).

- It results from the preceding assumption that, unlike the position adopted by Marx, constant capital (the means of production) does not include circulating capital, and is therefore constituted only of fixed capital. This makes it unnecessary to take into account a third section producing circulating capital. In addition to the resulting theoretical simplification, this is much more in line with the real world where final producers do not
advance circulating capital, but instead usually buy it with a credit from their suppliers of
intermediate goods.

- The economy is in a state of simple reproduction, from period to period, which in Marxist terms means that there is no accumulation of capital, whether it is constant or variable, from period to period, but a mere maintenance of the value of the existing stock of capital.

- To simplify the argument, it is assumed that the part of constant capital constituted by circulating capital is produced and incorporated into final goods in the same period in which it is produced. There is therefore no transmission of value from one period to the next by the part of constant capital constituted by raw materials or intermediate goods, in the absence of stocks of circulating capital.

- Finally the value produced in each period, which is equal to the quantity of living labor expended (directly or indirectly) during this same period to produce all of the final commodities, breaks down into L<sub>I</sub>, the value produced in section I, and L<sub>II</sub>, the value produced in section II.

Having reached the present stage, the reader who considers that the analysis of the simple reproduction carried out by Marx in Book II of Capital is correct, will certainly pretend that the model does not allow the analysis of the rate of profit, within the framework of these assumptions. In the case of simple reproduction, the value of the newly produced means of production is indeed not considered by Marx as surplus-value, since it merely replaces the value transmitted to the produced commodities by the existing stock of constant capital, for an identical amount. Surplus-value then exists only to the extent of the unproductive consumption of capitalists, which is impossible here, since it has been assumed that the capitalists do not consume. Surplus value and the rate of profit should therefore be zero.

In fact, we have already proved in previous chapters that fixed capital cannot transmit its value to the final product for many reasons. Let us just recall that in a theory of production as a transformation, the assumption of the transmission of value of constant capital is not legitimate if it applies to fixed capital. And constant capital is here constituted only of fixed capital, for the reasons already just explained above (cf. the second and third paragraphs of the previous page). It follows that Marx's reproduction schemes are erroneous. For the reader who would be interested in a full demonstration, based on the internal incoherence of these schemes, such a demonstration is provided in Appendix 2 at the end of the present book.

It will now be shown that the rejection of the assumption of transmission of value of fixed capital is perfectly compatible with the realization<sup>11</sup> of the product, whatever the type of reproduction envisaged, and this without falling into the type of contradictions that can be encountered in Marx's reproduction schemes. This approach will be explained, first for the realization of the product in Section II and then for the realization of the product in Section I.

<footnote n="11"><sup>11</sup> The word “realization” has the meaning of “sale”, but refers also to the realization of profits through the sale of the product, which explains that we prefer to use it rather than “sale”.</footnote>
## 3.2. The realization of the product and profits in Section II

In order for consumer goods to be realized, it is sufficient to assume that in exchange for the amount of monetary wages corresponding to the total newly produced value $L_I + L_{II}$, workers obtain consumer goods for a $L_{II}$ money value only, which implies that an $L_I$ value is thereby transferred without compensation to the capitalists of Section II who sell them these goods. This is achieved by assuming only that consumer goods are sold at an $L_I + L_{II}$ price higher than their $L_{II}$ monetary value, which allows capitalists of section II to levy an $L_I$ value equal to the total surplus value, but which corresponds now, after realization of the product in this consumer goods section, to the profits of this section alone. Let us call S this surplus value, and k the rate of surplus-value, by definition it is $k = \frac{S}{L_{II}}$. (1)

As in this very simple model $S = L_I = kL_{II}$ (by definition of k) the price of consumer goods, named $P_{II}$ must therefore be $P_{II} = L_{II} + kL_{II} = (1+k)L_{II}$ (2)

In this scheme, what is generally referred to as the margin rate in the macroeconomic sense in Section II, i.e. the ratio of profits named $\Pi_{II}$ over the price of consumption goods produced by section II, is thus equal to $\frac{\Pi_{II}}{P_{II}} = \frac{L_I}{L_I + L_{II}} = \frac{kL_{II}}{L_{II}(1+k)} = \frac{k}{1+k}$. (3)

This very simple mechanism just described is nothing else than the mechanism of surplus-value extraction, which is thus implemented through a necessary mechanism of transformation of money values into prices: in this simplified model it is sufficient that in exchange for their total monetary wages, corresponding to the production in money value of all the goods produced, workers are able to obtain only an $L_{II}$ money value, that of consumer goods only. This is so because, unlike values, which reflect the conditions of production, prices also reflect, beyond that, the conditions of distribution of the product between workers and capitalists, since they are the instrument of this distribution.

It should be noted that $w$ is the average monetary wage per unit of labor time. It has therefore a double dimension: that of money divided by the dimension of time. For simplicity, we can therefore assume that $w = 1$, but only if one takes into account the fact that $w = 1$ is a two-dimensional ratio (monetary unit/unit of labor time). In this case, as we have already explained at length, we may write $W = L_I + L_{II}$, but care must be taken that the quantities $L_I$ and $L_{II}$ are no longer average social values expressed in terms of labor time but are money values $wL_I$ and $wL_{II}$ expressed in monetary units that are pure scalars.

In any event, this makes it theoretically necessary to introduce the concept of value of labor-power as a concept that must be distinct from that of wages itself, since we need to express the difference between wages as the price of labor and the value of the goods obtained on the market in exchange for that monetary wage, which is the value of labor-power. In this simplified model, the overall value of labor-power (let us call it $V$) is thus equal to $L_{II}$.
The value of the labor-power in section II, let us call it $V_{II}$ corresponds to the share of the total value $L_{II}$ of consumer goods that the workers in section II obtain in exchange for their money wages (which amount to $W_{II} = L_{II}$). It is proportional to the share $\frac{L_{II}}{L_I + L_{II}}$ which is obtained by all workers in both sections in the total value of the whole product. This value $V_{II}$ is therefore equal to:

$$V_{II} = \frac{L_{II}}{L_I + L_{II}} \cdot L_{II} = \frac{L_{II}}{(1+k)L_{II}} \cdot L_{II} = \frac{L_{II}}{1+k} \qquad (4)$$

Since by definition $S = kV$ (we recall that $V$ is the value of the total labor-power), the surplus-value in this same section II, which we can call $S_{II}$ is thus $S_{II} = k \frac{L_{II}}{1+k}$. We verify that the ratio between the surplus-value and the value of labor-power $V_{II}$ in section II is equal to $k$. If we name $\Pi_{II}$ the monetary profits realized in section II, then the relation between these profits, which are equal to $L_I$ (in money value), and the labor price (money wages) in section II, which we have called $W_{II}$ (setting $w = 1$), is therefore equal to $\frac{L_I}{L_{II}} = k$: it is thus equivalent in section II to the rate of surplus-value.

### 3.3. The realization of the product and profits in Section I

What about section I, which produces fixed capital? As regards the value of labor-power in section I (let us name it $V_I$), it is for similar reasons equal to $\frac{L_I}{1+k}$. As it is known that by definition $L_I = k L_{II}$, it is easily verified that the sum of the values of labor-power of the two sections, which is $\frac{L_{II}}{1+k} + \frac{L_I}{1+k}$, is equal to $L_{II}$, the total value of labor-power. The surplus-value in section I is therefore $S_I = \frac{k}{1+k} L_I$, and since by definition we have $L_{II} = \frac{L_I}{k}$, the sum of the surplus-values of the two sections is:

$$S_I + S_{II} = \frac{k}{1+k} L_{II} + \frac{k}{1+k} L_I = \frac{k}{1+k} \cdot \frac{L_I}{k} + \frac{kL_I}{1+k} = \frac{L_I}{1+k} + \frac{kL_I}{1+k} = L_I \qquad (5)$$

With regard to the realization of the product in section I, we decide to call $a$ the fraction of the value of the total fixed capital that is accumulated (but not transferred to the product) in section I, and to call $b = 1 - a$ the fraction that is accumulated (but not transferred to the product) in section II.

Capitalists from section II will necessarily buy the fixed capital that they need to invest for maintaining their stock of capital from the capitalist of section I, who precisely produce this fixed capital. In order for these capitalists of section II to obtain in exchange for the money value $L_I$ thus levied on the workers only a fraction $bL_I$ of the money value $L_I$ of the newly
produced means of production, it is necessary and sufficient that the capitalists producing in section I the fixed capital used in the production of consumer goods, withdraw themselves, on the occasion of the sale of this fixed capital to capitalists of section II, the remaining fraction $aL_I$ of the value $L_I$.

Since it is trivial that $L_I = (1-a)L_I + aL_I$, this is what happens when these means of production to be used in section II, and of a value $bL_I = (1-a)L_I$ are sold at a price $L_I$, equal to $\frac{1}{b}bL_I$, or $\frac{1}{1-a}(1-a)L_I$. This price $L_I$ corresponds to the expense of the total profits of Section II, and allows the capitalists of section I who have insured this sale to levy in turn through this sale a profit equal to $aL_I$.

What remains to be done is to show how this remaining money value of the means of production needed for the production of fixed capital by section I is disposed of and realized. These means of production will be the object of successive purchases within Section I, going back upstream in the production of fixed capital. At this stage of our demonstration, it is appropriate to make an assumption about the share of the value of the available fixed capital used and purchased at each of these successive levels.

To be sure in the real world these shares are different from one level to the other. But taking that into account would introduce in our model a number of new variables, which would complicate unduly this attempt to shed some light on the realization of profits. In any case we know that these shares are necessarily always lower than one. Moreover the initial assumption has been made that capitalists in section I use a fraction $a$ of the total value of fixed capital produced globally by the productive system.

Therefore the simplest solution, to avoid the difficulty that we just mentioned, is to continue to assume that at each successive stage capitalists of section I always purchase the same fraction $a$ of the total money value of fixed capital still available after the previous purchases have been made. Let us consider it as a first assumption.

1) Capitalists in section I always purchase a fraction $a$ of the fixed capital available

It is on this basis that we shall now draw up a table, numbered 11.1. This table gives the prices, the money values realized and the money values levied as profits, on the occasion of each successive sale of fixed capital produced by section I. These variables are provided first for the sale to capitalists of section II, which will constitute the first row of our table, and then for the successive sales between capitalists of section I themselves, which constitute the following rows of this table 11.1.

As regards the first row, we have already seen above that by definition the value of the fixed capital obtained by the capitalists of section II and therefore the value of the fixed capital realized by those of section I by the corresponding sale, is a fraction $bL_I = (1-a)L_I$ of the total money value $L_I$ of this fixed capital, such realization being made at a price corresponding to $L_I$ and allowing the capitalists of section I who make this sale to make a profit. It is these three
elements which will appear in the table for this first row, knowing that the realized values appear there as a difference between the monetary prices and the profits, and as indicated above, express in fact the price of the corresponding labor, i.e. the amount of monetary wages.

In order to fill in the second row of the table, the rule should be used that, since we are here in a state of simple reproduction, all profits made at one stage on the sale of fixed capital are spent at the next stage for the purchase of fixed capital. It follows that the value levied as profits at one stage is identical to the price paid in the next stage. The other rule derives from the assumption made above that the purchase made at that price only permits to obtain a value equal to a fraction $a$ of the value $(1-b)L_t = aL_t$ of the fixed capital still available for section I. The value which is thus obtained is $a(aL_t) = a^2L_t$. Conversely, the price/value ratio is equal to $\frac{aL_t}{a^2L_t} = \frac{1}{a}$. The realized money value, in the form of wages, is therefore $a^2L_t$ and the value levied or profit is the difference between the price and the realized value, i.e.:

$$aL_t - a^2L_t = a(1-a)L_t \qquad (6)$$

At the next level, for the third row of the table, the price is therefore necessarily equal to $a(1-a)L_t$ (i.e. the profits from the previous stage), and the corresponding realized money value is therefore a fraction $a$ of this price, i.e. $a^2(1-a)L_t$. Since the remaining fixed capital has a value equal to $aL_t - a^2L_t = a(1-a)L_t$, we can verify that the realized money value is equal to a fraction $a$ of the value of this fixed capital still available at the end of the preceding stage. The profits levied at this stage amount to the difference between the price and the realized value, namely: $a(1-a)L_t - a^2(1-a)L_t = a(1-a)(1-a)L_t = a(1-a)^2L_t$.

The other rows of the table are obtained according to the same scheme, and therefore by recurrence. It is therefore unnecessary to explain them one by one. Let us note only that for row $n$ we get a price equal to $a(1-a)^nL_t$, which is the sum of the realized money value, i.e. $a^2(1-a)^nL_t$ and of profits equal to $a(1-a)^{n+1}L_t$. The complete table can now be given below:

<table><caption>**Table 11.1. Prices, money values and profits in section I (first case)**</caption><thead><tr><th>Prices</th><th>Realized money values = labor price = wages</th><th>Levied values = profits</th></tr></thead><tbody><tr><td>$L_t$</td><td>$bL_t = (1-a)L_t$</td><td>$aL_t$</td></tr><tr><td>$aL_t$</td><td>$a^2L_t$</td><td>$a(1-a)L_t$</td></tr><tr><td>$a(1-a)L_t$</td><td>$a^2(1-a)L_t = a^2L_t - a^3L_t$</td><td>$a(1-a)^2L_t$</td></tr><tr><td>$a(1-a)^2L_t$</td><td>$a^2(1-a)^2L_t$</td><td>$a(1-a)^3$</td></tr><tr><td>$\dots$</td><td>$\dots$</td><td>$\dots$</td></tr><tr><td>$a(1-a)^nL_t$</td><td>$a^2(1-a)^nL_t$</td><td>$a(1-a)^{n+1}$</td></tr></tbody></table>
We can verify that for each row of the above table we get the following identity:

Price = realized money value + levied value (profit).

The total price of the product for all the fixed capital sold by section I, named $P_I$, is the sum of the prices in the first column, namely:

$$ \Sigma \text{ prices} = P_I = L_I + aL_I + a(1-a)L_I + a(1-a)^2 L_I + \dots + a(1-a)^n L_I $$

$$ P_I = L_I + aL_I(1+(1-a)+(1-a)^2+\dots+(1-a)^n), \text{ which gives us when } n \to \infty: $$

$$ P_I = L_I \left( 1 + a \frac{1}{1-(1-a)} \right) = L_I \left( 1 + \frac{a}{1-a} \right) = L_I + L_I = 2L_I \quad (7) $$

It can be seen that the price of total fixed capital is twice the fixed capital price $L_I$ paid by section II, so that the fixed capital for section I (from the first row of the table) has the same price $L_I$ ! But this for a different realized money value: $(1-a)L_I$ for section II and $aL_I$ for section I.

The sum of the realized values for all the fixed capital produced from section I is in fact:

$$ \begin{aligned} \Sigma \text{ realized money values} &= (1-a)L_I + a^2 L_I + a(1-a)L_I + a^2(1-a)^2 L_I + \dots + a^2(1-a)^n L_I \\ &= (1-a)L_I + a^2 L_I (1+(1-a)+(1-a)^2+\dots+(1-a)^n) \end{aligned} $$

When $n \to \infty$, it is easy to check that this amount of $\Sigma$ realized money values is equal to:

$$ \Sigma \text{ money values} = (1-a)L_I + a^2 L_I \left( \frac{1}{1-(1-a)} \right) = (1-a)L_I + a^2 L_I \left( \frac{1}{a} \right) = (1-a)L_I + aL_I = L_I \quad (8) $$

It can be seen that the total amount $L_I$ of the money values realized by section I is equal to the sum of the value of the fixed capital sold to section II, that is $(1-a)L_I$, and of the value of the fixed capital sold within section I, or $aL_I$.

As for the sum of the levied values, i.e. the overall profit of section I, named $\Pi_I$, it is equal to:

$$ \Sigma \text{ profits} = \Pi_I = aL_I + a(1-a)L_I + a(1-a)^2 L_I + \dots + a(1-a)^n L_I $$

$$ \Pi_I = aL_I + a(1-a)L_I(1+(1-a)+(1-a)^2+\dots+(1-a)^n), $$

When $n \to \infty$, this gives us:

$$ \Pi_I = aL_I + a(1-a)L_I \left( \frac{1}{1-(1-a)} \right) = aL_I + a(1-a)L_I \frac{1}{a} = aL_I + (1-a)L_I = L_I \quad (9) $$
It can also easily be seen that the total profits of section I, $L_I$, is equal to the sum of profits realized in the sale of fixed capital to section II, i.e. $aL_I$, and profits realized in the sale of fixed capital within section I itself, i.e. $(1-a)L_I$.

It follows of course that the sum of prices in section I, i.e. $2L_I$, is equal to the sum of the realized money values $L_I$, the money value of the product of Section I, on the one hand, to which is added the sum of profits (the levied money values), or again $L_I$, on the other hand. To be sure these are very specific values, and we can legitimately question the reason for these particular figures. Here is the explanation: given that the price of the product in section I, i.e. $2L_I$, is equal by construction to $\frac{1}{1-\frac{1}{2}}L_I$, it follows that for section I as a whole value $\frac{1}{2}$

is the fraction of the value of the total available capital obtained on average through the overall spending of the profits of the two sections, which is $2L_I$. This is so because the sum of the parameters $a$ and $b$ is equal to 1, and therefore because their mean is equal to $\frac{1}{2}$. We may, therefore, ask ourselves what would be the case if another hypothesis had been made on the fraction of the available capital obtained at each stage by capitalists of section I. Suppose now, for example, that these capitalists of section I obtain at each stage not a fraction $a$, but as capitalists of section II, a fraction $b$ of the fixed capital still available. Let us consider it as a second assumption.

## 2) Capitalists in section I always purchase a fraction $b$ of the fixed capital available

This implies that the means of production are always sold at a price equal to $\frac{1}{b}$, i.e. $\frac{1}{1-a}$ times their value. This makes it possible to draw up a new table giving prices, realized money values and money values levied from each successive exchange of fixed capital between capitalists of section I. For the sale of fixed capital to capitalists of section II, which corresponds to the first row of the new table, there is obviously nothing changed compared to the previous table.

It remains to be seen how the remaining value $aL_I$ of the means of production utilized for the production of fixed capital is disposed of and realized. These means of production are procured in section I by going successively upstream in the production system. To make these successive purchases appear, we continue to assume that the means of production are sold at each stage at a price equal with this new assumption to $\frac{1}{b}$, or $\frac{1}{1-a}$ times their value. This makes it possible to draw up table 11.2, giving prices, realized values and values levied during each successive exchange of fixed capital between capitalists of section I.