<table><caption>Table 11.2. Prices, money values and profits in section I (second 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<sub>I</sub></td><td>bL<sub>I</sub> = (1-a)L<sub>I</sub></td><td>aL<sub>I</sub></td></tr><tr><td>aL<sub>I</sub></td><td>abL<sub>I</sub> = a(1-a)L<sub>I</sub></td><td>aL<sub>I</sub>(1-b) = a<sup>2</sup>L<sub>I</sub></td></tr><tr><td>a<sup>2</sup>L<sub>I</sub></td><td>a<sup>2</sup>bL<sub>I</sub> = a<sup>2</sup>(1-a)L<sub>I</sub></td><td>a<sup>2</sup>L<sub>I</sub>(1-b) = a<sup>3</sup>L<sub>I</sub></td></tr><tr><td>a<sup>3</sup>L<sub>I</sub></td><td>a<sup>3</sup>bL<sub>I</sub> = a<sup>3</sup>(1-a)L<sub>I</sub></td><td>a<sup>3</sup>L<sub>I</sub>(1-b) = a<sup>4</sup>L<sub>I</sub></td></tr><tr><td>...</td><td>...</td><td>...</td></tr><tr><td>a<sup>n</sup>L<sub>I</sub></td><td>a<sup>n</sup>bL<sub>I</sub> = a<sup>n</sup>(1-a)L<sub>I</sub></td><td>a<sup>n</sup>L<sub>I</sub>(1-b) = a<sup>n+1</sup>L<sub>I</sub></td></tr></tbody></table>

We can verify that for each line of table 11.2 we have: price = realized value + levied value (or profit).

The sum of prices, i.e. the price of the overall product of section I, named $P_I$, is equal to:

$$ \Sigma \text{ prices} = P_I = L_I(1+a+a^2+\dots+a^n) = L_I \left(\frac{1}{1-a}\right) \quad \text{when } n \to \infty \quad (10) $$

The sum of values obtained in section I is equal to:

$$ \Sigma \text{ realized values} = bL_I + abL_I + a^2bL_I + \dots + a^nL_I = bL_I(1+a+a^2+\dots+a^n) $$

When $n \to \infty$ this sum is equal to:

$$ \Sigma \text{ realized values} = \text{b. } \frac{1}{1-a} L_I, \text{ equivalent to } \frac{1-a}{1-a} L_I = L_I \quad (11) $$

As for the total of levied money values, i.e. the overall profits of section I, or $\Pi_I$, it is equal to:

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

$$ \Pi_I = aL_I(1+a+a^2+\dots+a^n) = L_I \frac{a}{1-a} \quad \text{when } n \to \infty \quad (12) $$

It follows, of course, that the sum of prices in section I is equal to the sum of realized values $L_I$, i.e. the money value of the product of section I, on the one hand, to which is added the sum of profits (levied values), on the other hand, since:

$$ P_I = L_I \frac{1}{1-a} = L_I + L_I \frac{a}{1-a} = L_I \left[ 1 + \frac{a}{1-a} \right] = L_I \frac{1}{1-a} \quad (13) $$

However this second case that we just examined may also be regarded as a special case with regard to the fraction of the available fixed capital purchased at each successive stage by capitalists of section I. It seems therefore necessary to examine a more general case, where
this fraction no longer takes values *a* (our first assumption) or *b* (our second assumption), but
a different but constant value. Let us consider it as a third assumption:

3) Capitalists in section I purchase a specified fraction *u* of the available fixed capital

Let us call *u* this value, with $0 < u < 1$, and draw up again the table of successive purchases of fixed capital, showing prices, realized values and levied values from each sale at each of the successive stages:

<table><caption>Table 11.3. Prices, money values and profits in section I (third case)</caption><thead><tr><th>Prices</th><th>Realized money values =<br>labor price = wages</th><th>Value of available<br>fixed capital</th><th>Levied values =<br>profits</th></tr></thead><tbody><tr><td>L<sub>i</sub></td><td>(1-a)L<sub>i</sub> = bL<sub>i</sub></td><td>L<sub>i</sub></td><td>aL<sub>i</sub></td></tr><tr><td>aL<sub>i</sub></td><td>auL<sub>i</sub></td><td>aL<sub>i</sub></td><td>a(1-u)L<sub>i</sub></td></tr><tr><td>a(1-u)L<sub>i</sub></td><td>au(1-u)L<sub>i</sub></td><td>a(1-u)L<sub>i</sub></td><td>a(1-u)<sup>2</sup>L<sub>i</sub></td></tr><tr><td>a(1-u)<sup>2</sup>L<sub>i</sub></td><td>au(1-u)<sup>2</sup>L<sub>i</sub></td><td>a(1-u)<sup>2</sup>L<sub>i</sub></td><td>a(1-u)<sup>3</sup>L<sub>i</sub></td></tr><tr><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>a(1-u)<sup>n</sup>L<sub>i</sub></td><td>au(1-u)<sup>n</sup>L<sub>i</sub></td><td>a(1-u)<sup>n</sup>L<sub>i</sub></td><td>a(1-u)<sup>n+1</sup>L<sub>i</sub></td></tr></tbody></table>

It appears that the gross profit margin for sales of fixed capital to capitalists of Section I, i.e. figures appearing from the second row of the table, is variable and equal to $1-u$.

It can be seen from the table that the sum of prices is equal to:

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

$$ P_I = L_I(1+a) + L_I a(1-u) \left[ 1 + (1-u) + (1-u)^2 + \dots + (1-u)^n \right], \text{ i.e. when } n \to \infty $$

$$ P_I = L_I \left[ (1+a) + a(1-u) \frac{1}{u} \right] = L_I \left[ (1+a) + \frac{a}{u} - a \right] = L_I \left( 1 + \frac{a}{u} \right) \quad (14) $$

The sum of realized money values in section I is equal to:

$$ \begin{aligned} \Sigma \text{ realized money values} &= bL_I + auL_I + au(1-u)L_I + au(1-u)^2 L_I + \dots + au(1-u)^n L_I \\ &= L_I \left[ (1-a+au) + au(1-u) \left[ 1 + (1-u) + (1-u)^2 + \dots + (1-u)^n \right] \right] \end{aligned} $$

Which gives us when $n \to \infty$:

$$ \Sigma \text{ realized money values} = L_I \left[ (1-a+au) + au(1-u) \frac{1}{u} \right] = L_I [(1-a+au)+a-au] = L_I (15) $$

Finally the sum of profits in section I, named $\Pi_I$, is equal to:
$$
\begin{align}
\Sigma \text{ profits} &= L_t \left[ (a+a(1-u)+a(1-u)^2+\dots+a(1-u)^n) \right] \nonumber \\
&= aL_t \left[ (1+(1-u)+(1-u)^2+\dots+(1-u)^n) \right], \text{i.e. when } n \to \infty: \nonumber \\
\Pi_t &= aL_t \left( \frac{1}{1-(1-u)} \right) = aL_t \frac{1}{u} = L_t \frac{a}{u} \tag{16}
\end{align}
$$

Here again, the sum of prices in section I is equal, on the one hand, to the sum of realized
values, $L_t$, i.e. the value of the product of section I (which, with $w = 1$, corresponds to the
price of labor in this section), to which is added the sum of profits (or levied values) of this
section, on the other hand, since:

$$
L_t \left( 1 + \frac{a}{u} \right) = L_t + L_t \frac{a}{u} = \Pi_{II} + \Pi_I = \Pi \quad (17)
$$

We can also verify that the two cases examined above in 1) and 2) are indeed particular cases
of this more general case, since the first case corresponds to a situation in which *u* = *a* and
the second case to a situation where *u* = *b* = 1 − *a*. To be sure, an even more general case
would be the situation prevailing in the real world, where at each stage of successive
purchases of fixed capital corresponds a distinct and different coefficient. Nevertheless, since
each of the coefficients *u* at each stage has a value such that 0 < *u* < 1, it is clear that the
corresponding numerical series is convergent. The economic interpretation of these results is
that the mechanism highlighted here is such that the whole value of the product of section I
can always be sold through this mechanism.

After seeing above in subsection 3.2. how the value produced in section II and the profits of
this section are realized, it has in any event been demonstrated in section 3.3. that all the value
produced in section I and the corresponding profits can be realized through a very simple
mechanism of successive purchases of fixed capital, and that the reproducibility of the system
is therefore ensured. This demonstration was conducted without having recourse to the
hypothesis of the transmission of value of fixed capital. It is thus verified at this level that this
value transmission is therefore not a necessary condition for the realization of the whole
product and for the reproduction of the productive system.

Moreover, it must be noted that the demonstration which has just been made is valid both for
simple reproduction and for extended reproduction, since no assumption has been made on
the value of the fixed capital used (in a final way) in each of the two sections of production:
this value indeed could have been equal to, or less than, or greater than the value of fixed
capital accumulated in the preceding period.

But this model, however simplified it may seem, and maybe because of its simple nature, has
a potential to teach us a few interesting lessons, as we shall try to demonstrate in the next
section.
## 4. Some characteristic quantities within the framework of the simplified model

At this stage, it is possible to calculate the values taken by some characteristic quantities in the framework of this simplified model. This will be done for the price of the global product, the ratio between wages and profits, and the investment rate, knowing that the calculation of the profit rate itself will only be addressed in the next chapter.

### 4.1. The price of the overall product

Taking up the three cases discussed in the previous chapter for the realization of the product in section I, we obtain the following prices for the overall product of the two sections (let us call Y this price):

1) In the first case, where $u = a$, we have:

$$Y = (L_I + L_{II}) + 2L_I = 3L_I + L_{II} = 3kL_{II} + L_{II} = L_{II}(1+3k) \quad (18)$$

$$Y = 3L_I + \frac{L_I}{k} = L_I \left( 3 + \frac{1}{k} \right) \quad (19)$$

2) In the second case, where $u = b = 1-a$, we have:

$$Y = (L_I + L_{II}) + L_I \left( \frac{1}{1-a} \right) = L_I \left( 1 + \frac{1}{1-a} \right) + \frac{L_I}{k} = L_I \left( \frac{2-a}{1-a} + \frac{1}{k} \right) \quad (20)$$

$$Y = L_{II}(1+k) + kL_{II}\left(\frac{1}{1-a}\right) = L_{II}\left(1+k+\frac{k}{1-a}\right) \quad (21)$$

3) In the third case, where u can be any number, we have:

$$Y = f(L_I) = (L_I + L_{II}) + L_I \left( 1 + \frac{a}{u} \right) = L_I \left( 2 + \frac{a}{u} \right) + \frac{L_I}{k} = L_I \left( 2 + \frac{1}{k} + \frac{a}{u} \right) \quad (22)$$

$$Y = f(L_{II}) = L_{II}(1+k) + kL_{II}\left(1+\frac{a}{u}\right) = L_{II}\left(1+2k+k\frac{a}{u}\right) \quad (23)$$

### 4.2. The profit/price of labor-power ratio

1) The profit/wages ratio in section II

In fact, it has already been pointed out above (see section 3.1 of this chapter) that in section II the ratio between profits, which may be called $\Pi_{II}$ and are equal to $L_I$, and the price of labor (wages), which may be called $W_{II}$, is equal to $\frac{\Pi_{II}}{W_{II}}$: it is therefore equivalent in section II to the rate of surplus-value. It follows in passing, which is not without interest, that from a statistical
and empirical point of view the ratio between wages and the overall price of consumer goods (a different ratio) is a good indicator of the rate of surplus-value existing in an economy, whatever it is.

2) The profit/wages ratio in section I

Using again the three cases examined in the preceding section regarding the realization of the product in section I, we obtain the following values for this ratio, which is $\frac{\Pi_I}{W_I}$:

- In the first case where $u = a$, we get:

$$ \frac{\Pi_I}{W_I} = \frac{L_I}{L_I} = 1, \text{ which shows well the peculiar nature of this case.} \qquad (24) $$

- In the second case where $u = b = 1 - a$, we get:

$$ \frac{\Pi_I}{W_I} = \frac{L_I \frac{a}{1-a}}{L_I} = \frac{a}{1-a} \qquad (25) $$

- In the third case (the general case) where $u$ can take any value, with $0 < u < 1$, we get:

$$ \frac{\Pi_I}{W_I} = \frac{L_I \frac{a}{u}}{L_I} = \frac{a}{u} \qquad (26) $$

Is striking to observe that for a given rate of surplus-value, which as it happens is $k$, there is a priori a notable difference between the profit/wages ratio in section II, producing consumption goods, where this ratio is always equal to $k$, i.e. the rate of surplus-value, and the ratio in section I, producing fixed capital, where it is independent from $k$, et depends in fact only on parameters $a$ and $u$.

However, if we have a look at the ratios which correspond to each stage of successive purchases of fixed capital in section I, they are of the general following form, for the $i^{th}$ stage of fixed capital spending within section I:

$$ \frac{\Pi_{I,i}}{W_{I,i}} = \frac{a(1-u)^{i+1} L_I}{au(1-u)^i L_I} = \frac{1-u}{u} \qquad (27) $$

$$ \text{From this last equation we can derive that for } u = a \text{ we have: } \frac{\Pi_{I,i}}{W_{I,i}} = \frac{1-a}{a} \qquad (28) $$

$$ \text{And that for } u = b = 1 - a, \text{ we get: } \frac{\Pi_{I,i}}{W_{I,i}} = \frac{1-(1-a)}{(1-a)} = \frac{a}{1-a} \qquad (29) $$
3) The profit/wages ratio for the system as a whole

$$- \text{ In the first case where } u=a, \text{ this ratio is: } \frac{\Pi}{W} = \frac{kL_t(a+u)}{uL_t(1+k)} = \frac{2ak}{a(1+k)} = \frac{2k}{1+k} \quad (30)$$

$$- \text{ In the second case where } u=b=1-a, \text{ it is: } \frac{\Pi}{W} = \frac{kL_t(a+1-a)}{(1-a)L_t(1+k)} = \frac{k}{(1-a)(1+k)} \quad (31)$$

$$- \text{ In the general case it becomes: } \frac{\Pi}{W} = \frac{L_I + L_I \left( \frac{a}{u} \right)}{L_I + L_{II}} = \frac{L_I \left( 1 + \frac{a}{u} \right)}{L_I \left( 1 + \frac{1}{k} \right)} = \frac{k(a+u)}{u(1+k)} \quad (32)$$

### 4.3. The I/Y investment rate and the share of profits in the product Π/Y

In the situation described by the simplified model under consideration, which is a simple reproduction situation where capitalists do not consume, it goes without saying that all profits are invested. As a result, the I/Y investment rate and the share of profits in the product, Π/Y, can only be identical.

Note also that if one reasoned purely in terms of values, the overall investment rate of the system, in a situation of simple reproduction, would be nothing other than the ratio between the value of the fixed capital newly produced and that of the global product, or:

$$\frac{I}{Y} = \frac{\Pi}{Y} (\text{values}) = \frac{L_I}{L_I + L_{II}} = \frac{kL_{II}}{L_{II}(1+k)} = \frac{k}{1+k} \quad (33)$$

The investment rate or the share of profits in section I alone would be equal to:

$$\frac{I_I}{Y_I} = \frac{\Pi_I}{Y_I} (\text{values}) = \frac{aL_I}{L_I} = a \quad (34)$$

As for the investment rate or the share of profits in Section II, they would be equal to:

$$\frac{I_{II}}{Y_{II}} = \frac{\Pi_{II}}{Y_{II}} (\text{values}) = \frac{bL_I}{L_{II}} = \frac{(1-a)kL_{II}}{L_{II}} = k(1-a) \quad (35)$$

If we now use the results in terms of prices obtained above, we get obviously slightly different expressions.

For the overall investment rate, or for the share of profits in the overall product, they are written in the form of the following ratio:
$$ 
\begin{gather*}
\frac{I}{Y} = \frac{\Pi}{Y}(\text{prices}) = \frac{L_I \left(1 + \frac{a}{u}\right)}{L_I \left(2 + \frac{1}{k} + \frac{a}{u}\right)} \\
\frac{I}{Y} = \frac{\Pi}{Y}(\text{prices}) = \frac{\frac{a+u}{u}}{\frac{k(2u+a)+u}{ku}} = \frac{k(a+u)}{k(a+u)+u(1+k)} \tag{36}
\end{gather*}
$$

In the case where $u = a$, this ratio becomes:

$$ \frac{I}{Y} = \frac{\Pi}{Y}(\text{prices}) = \frac{k(a+a)}{k(a+a)+a(1+k)} = \frac{2ak}{2ak+a(1+k)} = \frac{2k}{3k+1} \tag{37} $$

In the case where $u = b = 1-a$, the ratio changes as follows:

$$ 
\begin{gather*}
\frac{I}{Y} = \frac{\Pi}{Y}(\text{prices}) = \frac{k(a+1-a)}{k(a+1-a)+(1-a)(1+k)} = \frac{k}{k+1+k-a-ak} \\
\frac{I}{Y} = \frac{\Pi}{Y}(\text{prices}) = \frac{k}{k(2-a)+1-a} = \frac{k}{k+bk+b} \tag{38}
\end{gather*}
$$

As regards the investment rate or the profit share in Section I, which is simply expressed, they are equal to:

$$ \frac{I_I}{Y_I}(\text{prices}) = \frac{L_I \frac{a}{u}}{L_I \left(1 + \frac{a}{u}\right)} = \frac{\frac{a}{u}}{1 + \frac{a}{u}} = \frac{a}{a+u} \tag{39} $$

Finally, the investment rate or the share of profits in section II, which also have a simple expression, are equal to:

$$ \frac{I_{II}}{Y_{II}}(\text{prices}) = \frac{L_I}{L_I + L_{II}} = \frac{kL_{II}}{L_{II}(1+k)} = \frac{k}{1+k} \tag{40} $$

## 5. A few lessons from the simplified model

One could think that such a simplified model cannot teach us anything about the real world because its assumptions are precisely so simple that they do not correspond to this real world.

It is true indeed, for instance, that in the real world capitalists do consume, and in fact, on an individual basis, an average capitalist consumes most probably much more than an average worker. The assumption that there is no capitalist consumption in this simplified model seems therefore quite unrealistic. But going back to these assumptions, this model can nevertheless be rich in lessons.
5.1. The simplified model interpreted as excluding capitalists

Firstly, since there is no capitalist consumption in this model, one might imagine that we
could go one step further and consider that there are no capitalists at all in the model. Then
these assumptions would correspond to a world where there would be no private property of
fixed capital, meaning that this capital would be the collective property of all workers, who
would ipso facto be the owners of all the commodities which are produced, be they
consumption goods or fixed capital goods. In such a situation, there would be no surplus-
value, and one could think that there would be therefore no need for a difference between the
money value and the price of commodities, since it is this difference which allows for the levy
of surplus-value under the form of profit. In this interpretation, each worker would obtain a
value equivalent to that of the commodities which it has contributed to produce, in proportion
to its contribution to production, measured by the level of his wage.

This idea would nevertheless be wrong, since what ultimately production is destined to satisfy
is consumption, and since fixed capital can be accumulated (or even contemplated!) but by
definition cannot in any case be consumed like a consumption good, which means that
workers of section I producing fixed capital cannot be interested in the property of these
goods that they produce, or of their value, but quite obviously are rather interested, like
workers in section II, in the acquisition of consumer goods. This reminds us that fixed capital,
considered as a whole, is produced only to be employed – ultimately, in the production of
consumption goods. We go back there to the old conception, dating back to Böhm-Bawerk
and the Austrian school, of capital viewed as a production detour, a process also called
“roundaboutness”.

This implies that no more than capitalists, workers in section I producing fixed capital goods
cannot be satisfied to obtain the very goods that they have produced, for a value
corresponding to the average social labor which they have spent to produce these goods. Since
- under the particular assumption of this subsection, there is no private property of fixed
capital, they might be considered as the collective owners of all this fixed capital. But even
though a worker in section I might thus be considered as the owner of a fraction of fixed
capital which he has contributed to produce (to the tune of the amount of his wage), this
would not provide him with the consumption goods he needs, in order to make a living.

It results from this recognition that all workers of both sections need to buy all of the
consumption goods which have been produced by workers of section II only, and that the
overall price of these goods must necessarily be equal to the overall amount of wages, which
is higher (by an amount $L_I$ or $W_I$) than their value $L_{II}$ or $W_{II}$. This in turn implies that whatever
the mode of production, the existence of fixed capital explains and implies the necessity for
prices of consumption goods to be higher than their money value.

Secondly, this also implies the existence of money as a general medium of exchange, which is
the only way to avoid that workers of section I enter into some kind of weird exchanges of
fixed capital goods against consumption goods: one cannot see how an economic system
would work with such a kind of barter trade, although such a system constitutes the basis of
neoclassical economics! Here we can see that, through its role as a general medium of exchange, money plays a very important role in allowing for the distribution of the product of the system to take place. In the present case this is the distribution of consumption goods among workers: the fact that wages are paid in money allows workers of section I to share without any difficulty the consumption goods produced in section II with workers of this last section. This distribution function of money plays such an essential role that it might be worth considering it as a fourth function of money.

Thirdly, and as a consequence of what precedes, one must realize that, by interpreting the assumptions of this one-step-further model (without private property of fixed capital) as meaning that there are no capitalists in this system, we arrive at a situation where there are also no true profits, because there is no need for them: profits would materialize only in case of private property of fixed capital, i.e. if there were no collective property of this kind of goods.

To continue with the examination of the operation of this model without capitalists, we can imagine a way to materialize the fact that workers of both sections, once having shared all of the consumption goods produced, would also be the collective owners of the fixed capital produced in section I. As we have seen previously, since the price of all consumption goods is equal to their value $L_{II}$ multiplied by $1 + k$, the simplest way would be to deliver to the buyer of any consumption good a certificate representing the ownership of fixed capital, for an amount equivalent to the price of this good multiplied by $\frac{k}{1+k}$. Even more simply, the receipt delivered for each purchase could mention this amount, for instance together with the amount of value-added tax (VAT) when such a tax exists: this VAT might itself be seen as a kind of collective contribution to the State budget, involving the right to enjoy public services provided by it.

It would nevertheless be understood that a certificate issued on the occasion of the purchase of whatever consumption good would not be exchangeable against money or any other good. Indeed it would just be a symbol of sharing the collective property of the new fixed capital produced during the period during which this consumption good has also been produced. Moreover, because there are no capitalists in this system, as we already pointed out, there would also be no profits, which means that the property of fixed capital would not give the right to sell this type of good and more importantly the right to derive a profit from its ownership. But, apart from bare ownership, it would have obviously kept its even more important feature, i.e. the right to use fixed capital, in the present case for the production of commodities (we shall come back below to these various attributes of property).

## 5.2. The simplified model and money creation

An other important characteristic of this simplified model has to do with money creation. Continuing to assume that there are no capitalists and that at each period production starts from scratch, this production could not be sold if the corresponding wages had not been paid previously, and wages could not be paid if money had not been obtained from the institutions
which create it, i.e. banks: in other words the public entity in charge of managing the
productive system needs to borrow from them in order to pay for these wages. The amount of
money which thus needs to be borrowed, in order to be injected in the circuit through the
payment of wages, corresponds precisely to the overall amount of these wages, i.e. $W_I + W_{II} = L_I + L_{II}$ (with $\bar{w}=1$). As for the repayment of this amount, it poses no problem as long as
the collective property of fixed capital allows to consider the whole productive system as a
unique giant integrated enterprise, owning both sections of production.

However, if each one of the two sections had a kind of legal autonomy and distinct legal
personality, meaning that the collective property of commodities would be split between two
sets of workers with two separate entities managing each section (in accordance with the
structuration of the productive system between these two sections), there would undoubtedly
be a problem, because a distinct amount of money $W_I = L_I$ would have been borrowed by
section I, but would accrue to section II through the purchase of consumption goods by
workers of section I. In order to get this amount back and repay the borrowed amount of
money, section I would thus have to sell fixed capital to section II for a similar amount $W_I$.
Then $W_I$ would appear as a kind of virtual profit, corresponding exactly to the sale of fixed
capital, for the part of it needed by section II, and which would allow for the repayment of the
same amount of money initially borrowed in order to pay for the wages of section I.

This shows us that it is the dissociation of property (be it collective or not) which creates the
logical need for a transfer of income from wages, which are the original or primary income. It
equally implies the differentiation of price and value also in section I, and the need for the
price of fixed capital to be higher than its value, like in the case of consumption goods. Indeed
the price $W_I = L_I$ of the fixed capital goods bought by section II would necessarily be higher
than their value, since these fixed capital goods would represent not the whole production of
fixed capital, but only the part $(1-a)L_I$ of fixed capital needed by and used in section II, and
not the remaining part $aL_I$ of fixed capital needed by section I itself. The fixed capital
corresponding to this last part would not be sold: because it has only one collective owner, it
would stay in section I where it has been produced, and therefore neither a particular price nor
further virtual profits would be involved in its distribution.

However, if section I itself were divided into several legal entities corresponding to the
technical structuration of the production process between the different kinds of fixed capital
needed at each of its stages, then transactions to sell and buy this fixed capital between these
separate entities would have to take place, and would involve new amounts of virtual profits.
It is this process that we have described earlier in this chapter, when we examined the
realization of profits in section I. We showed above (see equation (16) p. 213) that in this case
the total amount of (virtual) profits in section I would be $\Pi_I = L_I \frac{a}{u}$.
## 5.3. Surplus-value and monetary profits

The simplified model exposed in this chapter shows therefore that whatever the type of property of fixed capital (individual or collective) it is the existence of multiple entities, each being entitled to a property right on its fixed capital and its products - which logically implies the existence of successive transfers of income originating from wages. Moreover these entities can be linked or not to the technical structuration of the productive system, because there can be several entities for a particular technical stage, but this technical aspect is not what only matters. Incomes resulting from transfers occurring between these entities are logically called profits, since their origin is linked to the sale and purchase of commodities through monetary prices higher than their values, even though profits in this simplified model are to some extent somewhat virtual, as long as it is interpreted as a model without capitalists.

Indeed, if we abandon the idea of a collective property, and go back to the case of individual property, implying that we reinstate the existence of capitalists, let us just consider that this property covers the right to perceive money profits from the sale of products: we shall go back in the next subsection on the various attributes of property. Nevertheless, and by construction in this model, profits cannot be used to buy consumption goods, although this is clearly the ultimate goal of any normal individual, even he is a capitalist. As for the amount of these profits, apart from profits made through the sale of consumption goods (which in the simplified model only correspond to purchases made from wages), this amount depends entirely only on the structuration of the productive system in section I: it is in fact the degree of integration of production in this section (measured by the degree of property fragmentation) which determines its precise amount.

In other words, for a given amount of surplus value and profits in section II, realized in this section when workers buy consumption goods, the amount of monetary profits depends on the structure of property in section I, which is overlapping on the technical structure of production in this section. It is indeed easy to understand that even in a capitalist system, if the fixed capital of section I is entirely produced by only one huge capitalist enterprise, then the total amount of monetary profits for the whole productive system will simply be $L_I = kL_{II}$, i.e. only monetary profits obtained in section II, because in section I the income $L_I = W_I$ earned from the sale of fixed capital to section II at this price $L_I$ will be totally offset by the repayment of the total amount of wages which have been paid in section I.

On the contrary, if capitalist property in section I is split among at least as many private companies as there are stages in the production process in this section, then the amount of profits in this section will be $\Pi_I = L_I \frac{a}{u}$, as indicated above.

## 5.4. The simplified model and profits as a transfer income

Finally, this demonstration sheds some light on a mechanism which is at work behind the formation of all incomes created through a transfer occurring in the sphere of circulation, and
by which the expense of an income creates a new income. We will now expand a little on this
interesting mechanism, as far as profits are concerned.

It should first be noted that, as regards the realization of fixed capital produced in section I, fixed capital exchanges are "successive" only for the sake of clarity. This succession is logical and not chronological, because these exchanges can perfectly be simultaneous. There is no need to refer to a particular period or to successive periods, in particular because of the existence of money and credit: during each period, however short it may be, and in fact at any moment, capitalists of section I sell the needed quantity of fixed capital to those of section II, and sell to one another in section I the quantity of fixed capital corresponding to their own needs. It is important to note that it is this mechanism based on the spending of profits that does make it possible to produce fixed capital goods in section I.

In this mechanism of successive spending of profits in section I that was exposed above in
section 3, we thus find, but with a different method, a result which had been well brought to
light by Bernard Schmitt, in his book “Monnaie, salaires et profits” (“Money, wages and
profits”), which we cited previously, on the basis of a theory of money and monetary circuit
in which profit is a transfer income. However Schmitt’s theory does not explain the modalities
of this transfer through a reproduction scheme, as we did, for it is an axiomatic theory of
money, founded on its circuit, and not a complete theory of production and distribution. In the
model developed here, on the contrary, we see clearly that profits are the result of a double
transfer:

* A first transfer, which takes place on the occasion of the expenditure of wages by workers of the two sections of production, for the purchase of consumer goods from capitalists of section II. It is through this first transfer that surplus-value is levied, once and for all, under the form of monetary profits, and that its amount is fixed, and is no longer liable to change. In this simplified model, this transfer simultaneously sets the amount $L_I$ of monetary profits for capitalists of section II.

* A second transfer, of a very different nature, since it only involves capitalists, and which takes place on the occasion of the purchase by capitalists of section II, from those of section I, of the fixed capital that they need to produce consumer goods. This purchase has no influence on the amount of surplus-value, which has already been determined, but it fixes the distribution of this total surplus-value $L_I$ between capitalists of the two sections. The share accruing to capitalists of section II is $(1-a)L_I = bL_I$, as we saw in the previous section, and that accruing to the capitalists of section I is $aL_I$, which also corresponds to the amount of monetary profits going to the capitalists who ensured this sale. This first transfer to section I will then be followed by (logically) successive transfers within section I, transfers which will ensure the distribution of this surplus-value $aL_I$ and of the corresponding profits between the capitalists of this same section.

These observations provide us with the opportunity to go a little further as regards the
difference between primary incomes and secondary or transfer incomes:
As regards primary incomes, they are such because they are paid and earned on the occasion
of the production of commodities, which logically precedes their circulation. In our model
where all labor is made of wage-labor, wages are for this reason the only primary incomes. It
is so because the payment of wages is indeed a necessary and sufficient condition for
production to take place. A necessary condition, because no production can be carried out if
workers are not previously hired and their wages specified: these wages are earned at the
same pace as production and paid according to a specified periodicity. Thus, any amount of
wages corresponds to a period during which work has been performed and production has
been carried out. We can note in passing that this period can sometimes be even shorter than
the time needed for the resulting commodities to be brought to the market.

The payment of wages is also a sufficient condition for production to take place because once
workers are hired and at work, no distribution of (new) profit is needed for production to be
carried out: this might seem difficult to understand in a disaggregated economy, but becomes
obvious if we imagine that the productive system is made of a single fully integrated firm,
producing both consumption goods and all of the intermediate and fixed capital goods needed
for the production of the former. This means that no purchases involving the payment of
profits have to be made by this single firm from other firms. This observation confirms and
reinforces the conclusion that we had already reached above (see subsection 6.4. p. 191), and
which is the nullity of profits at the stage of production.

Finally, wages are linked to production by their very nature: as we have seen they have the
dimension of a scalar quantity per unit of time, whereas the value of what is produced has
itself the dimension of time: it is the payment of wages which transforms this value in money
value and reciprocally, by a process of codetermination.

Conversely, as regards profits, they can only form at the level of distribution of the product, if
there is a first money transfer from workers to capitalists of section II, when these workers
spend their wages to buy consumption goods, and this at a monetary price which is
necessarily higher than the money value of these same goods. As for profits realized by
capitalists of section I producing fixed capital goods, they will never be realized if there is not
a second money transfer, i.e. if capitalists of section II do not invest in the purchase of fixed
capital.

This mechanism of realization of the product and profits in section I brings us back to a
principle first highlighted by Keynes, who introduced it in his “Treatise on Money”, published
in 1930, six years before his “General Theory…”. It is the principle of the "widow's cruse",
according to which it is the expense of profits which creates profits. For Keynes, however,
this principle must have not seemed so important because he writes: “There is one peculiarity
of profits (or losses) that we may note in *passing*, because it is one of the reasons to segregate
them from income proper, as a category apart. If entrepreneurs choose to spend a portion of
their profits on consumption [...] the effect is to increase the profit on the sale of liquid
consumption goods by an amount exactly equal to the amount of profits which have been thus
expended [...] Thus, however, much of profits entrepreneurs spend on consumption, the
increment of wealth belonging to the entrepreneurs remains the same as before. Thus, profits,
as a source of capital increment for entrepreneurs, are a widow's cruse which remains un-
depleted, however much of them may be devoted to riotous living" (Keynes, 1930, chapter 10,
p. 139).

However our simplified model shows that the mechanism that Keynes considered as a
"peculiarity" is not such a particular device and must on the contrary be generalized to explain
the realization of profits, which at the same time sheds lights on their nature as a transfer
income. Moreover, when Keynes described this - in fact fundamental - mechanism of the
widow's cruse, he was referring to the spending of profits to buy consumption goods, whereas
we have demonstrated that this mechanism is also at work even when there is no capitalist
consumption. In any case Keynes drew our attention to what we think is a neglected but
similarly fundamental element of the system: the consumption of capitalists. This element is
in our view of such importance that we will now devote to it the full chapter which follows.

However, before addressing this question, we can go a little further in the understanding of
profits, by asking the question of their nature, i.e. the reasons that make it possible for
capitalists to realize them through a transfer from primary incomes. The answer is quite
simple, because it is clear that this capacity to realize and keep profits is due to the capitalists'
ability to fix prices at a level which is higher than wages, i.e. the primary incomes which they
have payed to the workers that they employ, such a level that it allows them to levy their
profits out of the amount of their sales. This capacity to fix prices derives itself from the fact
that capitalists are the owners of the commodities which they sell on the market.

We are here at the heart of the system, this heart being the social institution of property, and
more precisely private property. This fundamental institution is defined by law, and in
capitalist societies it has kept all the main characteristics that it had under Roman law,
according to which property rights are divided in three categories: usus, fructus and abusus.

- Usus contains the rights to use the thing that is owned, such as inhabiting a house or an apartment. In the case of the capitalist system property of fixed capital gives the right to put it at work with hired workers.

- Fructus contains the right regarding the products of a property, such as the fruits of a tree or the crops gathered from a piece of land, or in the capitalist system the commodities produced by any production process involving fixed capital, because this capital, as much as labor, is supposed to be productive in itself.

- Abusus contains the right to dispose of a property, such as selling, modifying, or even destroying it, which allows capitalists to sell obviously fixed capital itself, but also all commodities coming from the production process, since - as a part of fructus, they are the property of the producer.

All incomes are created as money incomes, to begin with wages, and all transfer incomes exist because they result from a transfer of property, i.e. the transfer of the property of some commodity from the seller to the buyer in exchange for money. It is important to understand that such a transfer can occur only because the commodity which is transferred pre-exists - as
the property of the seller - to this exchange against a specified amount of money, this amount
being the monetary price of the transferred commodity. In fact there would not be any
exchange if property did not pre-exist: property is a precondition of exchange. This explains
that in the neoclassical fairy tale, there must be an endowment of commodities to individuals,
before they go to the market, even though no production has taken place previously, a
situation which defies the most basic logic.

This has a very profound consequence, because if an exchanged commodity already exists, it
means ipso facto that it has already been produced previously, and that these exchanges by
themselves do not create anything or any additional value. This explains why these transferred
incomes cannot produce themselves anything or create any value, since value already exists: it
has been created earlier as a dimension of commodities resulting from the production process
itself. Therefore these exchanges cannot do anything else than shifting the ownership of pre-
existing commodities by moving them from the seller to the buyer in opposite directions to
pre-existing monetary income. This is the reason why these transferred incomes cannot
therefore be anything else than a redistribution of pre-existing primary incomes, i.e. wages in
this simplified model.

If we took into account what Engels called “simple commodity production”, then incomes
from self-employed producers, corresponding to the sale of commodities just produced from
their own labor, should also be considered as primary incomes. We concur indeed with
Engels, who writes: “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” (Engels, 1894,
supplement to Capital, Vol. 3, Law of value and rate of profit, on-line version).

It follows that apart from wages, which are revenues paid for buying workers labor, which
allows to use this labor in the production process, or incomes of self-employed producers, all
other revenues appear in the sphere of circulation, where all exchanges of commodities take
place, and are revenues from property, be they profits from the property of fixed capital, rents
from the property of housing, interests from the property of money assets, or amounts of
money earned from the sale of existing assets (real ones, like lands, real estate, second-hand
goods, or financial ones, like companies shares or bonds).

At the end of this chapter, it is now time to summarize the main lessons through some
principles that we can derive from all these observations, firstly on surplus-value, and
secondly on incomes.

**Principle 19:** In the capitalist mode of production, fixed capital is by definition the
property of capitalist but not of workers, who own neither the means of production nor
the commodities which they create through their labor. The simple reproduction of this
system as it is implies therefore that workers cannot get on the market, from the
spending of their wages, the property of at least the part of the product consisting of
means of production. The difference between the whole product of labor and the share
obtained by workers is defined as surplus-value.
**Principle 20:** Surplus-value is created in the sphere of production, but extracted in the sphere of circulation, where commodities are distributed among various stakeholders. This results from the fact that wages, as the money price of labor-power representing the money value of the whole product, can buy only a share of this product consisting in consumption goods, whose price is always higher than their money value. Surplus-value thus materializes under its monetary form as profits in the section producing consumption goods. These profits are then distributed among capitalists.

**Principle 21:** The only primary incomes are the revenues created on the occasion of the production process, and are the monetary counterpart or remuneration of labor, i.e. wages in a pure capitalist mode of production.

**Principle 22:** Profits, as any income other than wages or the remuneration of labor, derive from the property of commodities or assets. They are transfer revenues which arise from exchanges against money of previously produced commodities (or assets) taking place in the circulation process.

## References

Althusser, Louis (1965). *Pour Marx*. Paris: François Maspero. English Edition: Althusser, L. (1969). For Marx. (translation: Brewster, B.). London: Allen Lane, the Penguin Press.

Althusser, Louis & Balibar, Etienne (1965). *Lire le Capital*. Paris: François Maspero. English Edition: Althusser, L. & Balibar, E. (1970). *Reading Capital* (translation: Brewster, B.). New York: Pantheon Books, Random House.

Bortkiewicz (Von), Ladislaus (1907). On the Correction of Marx's Fundamental Theoretical Construction in the Third Volume of Capital. Originally published in *Jahrbücher für Nationalökonomie und Statistik*. Vol. XXXIII (N°3, July 1907). English edition and translation by Paul M. Sweezy in Böhm-Bawerk (1949). *Karl Marx and the Close of his System* (pp.198-221). New-York: Sweezy Paul M. (ed.), Augustus M. Kelley.

Engels, Friedrich (1894). A supplement to Capital Volume Three. In 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 16 Aug 2018.

Flamant, Christian (2015). 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. *World Economic Review*, issue N° 5 (July 2015), pp. 22-39.

Keynes, John Maynard (1930). *A Treatise on Money*. Last edition: 1971. London: Macmillan.

Marx, Karl (1885). *Capital: A Critique of Political Economy. Volume II - Book One: The Process of Circulation of Capital*. Edited by Friedrich Engels. Source: First English Edition of 1907. Marx/Engels Archive [online]. Available at:
https://www.marxists.org/archive/marx/works/download/pdf/Capital-Volume-II.pdf
Accessed 25Aug 2018

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

Pasinetti, Luigi (1981). *Structural Change and Economic Growth – A theoretical essay on the dynamics of the wealth of Nations*. Cambridge (UK): Cambridge University Press,.

Samuelson, Paul (1971). Understanding the Marxian Notion of Exploitation: A Summary of the So-Called Transformation Problem Between Marxian Values and Competitive Prices. *Journal of Economic Literature*, Vol. 9 (No. 2), pp. 399-431.

Schmitt, Bernard (1966). *Monnaie, salaires et profits (Money, wages and profits)*. Paris: PUF.

Sraffa, Piero (1960). *Production of commodities by means of commodities*. *Prelude to a critique of economic theory*. Cambridge (UK): Cambridge University Press.
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# Chapter 12. A more realistic model integrating capitalist consumption

In this new model it will be shown that taking into account capitalist consumption not only brings about a much more realistic view of the functioning of the capitalist system, but also sheds light on a very important element of this system as far as production and distribution are concerned. This will be done by exploring first the consequences on some key variables of this introduction of capitalist consumption, as a new and fundamental element of the model. The implications of this important modification on the realization of the product and profits in both sections of the productive system will be discussed as a second step.

## 1. The main changes to the initial simple model

Let us first introduce the consumption of capitalists, that we call $C_c$, in a system which is still in a state of simple reproduction. This consumption is of course and by nature made only by capitalists as individuals, since they benefit in one way or another (dividends, etc.) from the distribution of profits made by firms (because these firms by definition do not buy any consumption goods). We assume that this consumption represents a constant fraction, called $c$, of the $L_{II}$ value of consumer goods produced in each period, with ($0 < c < 1$, and $\bar{w}=1$).

We thus have $c = \frac{C_c}{L_{II}}$ and therefore capitalist consumption is by definition: $C_c = c L_{II}$ (1)

What then becomes the system of values? In this context, the value of the product of each section (let us name these products $Y_I$ and $Y_{II}$ respectively) remains unchanged at $L_I$ for section I, $L_{II}$ for section II and $L_I + L_{II}$ for $Y$, i.e. the total product. Moreover, by definition, the amount of the total surplus-value $S$ levied on workers increases correspondingly from $L_I$ to $L_I + cL_{II}$, which is equivalent to $k L_{II} + c L_{II}$, i.e. $L_{II} (k + c)$, and the value of the total labor-power decreases in proportion as it passes from $L_{II}$ to $L_{II} (1 - c)$. As for the value of the accumulated capital at each period $I_t$ (simply called $I$ in a situation of simple reproduction) it is still equal to $L_I$. Let us also call $S = L_{II} (k + c)$ the new surplus-value resulting from these new values, and $V$ the new value of the labor-power which corresponds to it. Let us call $k_c$ the new rate of surplus value, which is thus written:

$$k_c = \frac{S}{V} = \frac{L_{II}(k+c)}{L_{II}(1-c)} = \frac{k+c}{1-c} \qquad (2)$$

$$\Rightarrow S = V k_c = V \frac{k+c}{1-c} \qquad (3)$$

Knowing that the rate of surplus-value $k_c$ is by definition the same in the two sections, and that the value of the product of each section is by definition equal to the value of labor-power
plus the surplus-value, $L_I = V_I (1+k_c)$ and $L_{II} = V_{II} (1+k_c)$. The value of labor-power is immediately derived from that:

$$ \bullet \text{ in section I: } V_I = \frac{L_I}{1+k_c} = \frac{L_I}{1+\frac{k+c}{1-c}} = \frac{L_I}{1+k} = \frac{L_I(1-c)}{1+k} \quad (4) $$

$$ \bullet \text{ and from a similar calculation in section II: } V_{II} = \frac{L_{II}}{1+k_c} = \frac{L_{II}(1-c)}{1+k} \quad (5) $$

It can easily be verified that the sum of these two values is equal to the overall value V of labor-power:

$$ V_I + V_{II} = \frac{L_I}{1+k_c} + \frac{L_{II}}{1+k_c} = \frac{kL_{II}}{1+\frac{k+c}{1-c}} + \frac{L_{II}}{1+\frac{k+c}{1-c}} = \frac{L_{II}(1+k)}{1+c} = L_{II}(1-c) = V \quad (6) $$

Surplus-values $S_I$ and $S_{II}$ in each section are also easily obtained, since it is by definition equal to the value of labor-power multiplied by the rate of surplus-value $k_c$ (see equation 2).

$$ \text{In section I, surplus-value } S_I \text{ is therefore equal to: } k_c \frac{L_I}{1+k_c} = \frac{L_I \frac{k+c}{1-c}}{1+\frac{k+c}{1-c}} = L_I \frac{k+c}{1+k} \quad (7) $$

$$ \text{In section II, the calculation is similar, with } S_{II} \text{ equal to: } k_c \frac{L_{II}}{1+k_c} = L_{II} \frac{k+c}{1+k} \quad (8) $$

It can also be verified that their sum is equal to total surplus-value, as previously defined:

$$ S = S_I + S_{II} = L_I \frac{k+c}{1+k} + L_{II} \frac{k+c}{1+k} = kL_{II} \frac{k+c}{1+k} + L_{II} \frac{k+c}{1+k} = (k+1)L_{II} \frac{k+c}{1+k} = L_{II}(k+c) \quad (9) $$

# 2. Realization of the product and profits in a model with capitalists consumption

## 2.1. Realization of consumption goods and profits in section II

As we saw previously, for the realization of the product to take place, the level of prices must be such as to allow this realization. The difference that is now introduced, in comparison to the simple model previously exposed, is that the realization of consumer goods of a money value $L_{II}$ must now take into account the consumption of capitalists, and therefore the fact that workers can no longer buy the totality, but only a fraction $(1-c)$ of this value $L_{II}$.

Let us first recall that in the simplified model without capitalist consumption, workers produce final goods with an $L_I + L_{II}$ value, for which they receive, in monetary units, wages amounting to $W = L_I + L_{II}$ (with $\bar{w}=1$). Then when they spend these wages they obtain in return consumer goods for only an $L_{II}$ value, corresponding to the value of labor-power, since in a simple reproduction situation the price of these consumption goods is fixed at $L_I + L_{II}$, or $L_{II} (1+k)$ (see above). This is the simple mechanism by which surplus-value is levied. The
profit realized in this section is then $L_I = kL_{II}$. By calling $s_{II}$ the ratio between these profits in section II and the price of labor in the same section, it is, as we have seen, identical to the rate of total surplus-value:

$$s_{II} = \frac{kL_{II}}{L_{II}} = k \qquad (10)$$

In the present and more complex model of simple reproduction with capitalist consumption, the reduction in the value of workers consumption to (1 - c) $L_{II}$ does not in any way imply a fall in the price of labor, which remains fixed at $L_I + L_{II} = (1 + k) L_{II}$ (i.e. the amount of wages with $\bar{w} = 1$).

In addition, we know that:
$$\frac{1}{1-c} = \frac{c}{1-c} + \frac{1-c}{1-c} = \frac{c}{1-c} + 1 \Rightarrow \frac{c}{1-c} = \frac{1}{1-c} - 1$$

Moreover the rate of surplus value $k_c$ can be expressed as
$$k_c = \frac{(1+k)-(1-c)}{1-c} = \frac{1+k}{1-c} - 1$$

$$\text{This gives us conversely: } \frac{1+k}{1-c} = 1+k_c \qquad (11)$$

For these wages which stay at an amount of $(1 + k) L_{II}$ to buy only a fraction (1 - c) of the consumer goods, it is therefore necessary and sufficient that consumer goods of a money value equal to $L_{II}$ be sold at a price now increased in such a way that it passes from:

$W = P_{II} = (1 + k) L_{II}$, as they were in the previous and simple model, to:

$$P_{II} = \frac{1}{1-c} P_{II} = L_{II} \frac{1+k}{1-c}, \text{ or } P_{II} = L_{II} (1+k_c), \qquad (12)$$

This price $P_{II}$ is such that it gives the possibility to realize in this section II an additional monetary profit equal to this capitalist consumption, i.e.:

$$C_c = cL_{II} \cdot \frac{1+k}{1-c} = cL_{II} (1+k_c) \qquad (13)$$

It is indeed easy to verify that this price $P_{II}$ makes it possible to realize the total $L_{II}$ value of consumer goods, since by adding the consumption of workers $C_w = W$ and that of capitalists, one obtains the overall price of consumer goods. Indeed, since $W = L_I + L_{II} = L_{II} (1 + k)$, we can see immediately that:

$$C_w + C_c = W + Cc = L_{II} (1 + k) + c L_{II} \cdot \frac{1+k}{1-c}, \text{ which is equal to:}$$
$$W + Cc = (1-c) L_{II} \cdot \frac{1+k}{1-c} + c \cdot L_{II} \cdot \frac{1+k}{1-c} = L_{II} \cdot \frac{1+k}{1-c} = L_{II} \left( \frac{k+c}{1-c} + 1 \right) = L_{II} (1+k_c) = P_{II} \quad (14)$$

From this last equation, we can consequently derive that workers obtain a fraction $(1-c)$ $L_{II}$ of the (money) value $L_{II}$ of consumer goods whose overall price is $P_{II} = L_{II} \cdot \frac{1+k}{1-c}$, with capitalists of section II obtaining the remaining $c$ $L_{II}$ fraction of this same value.

The price for these workers' consumption goods is: $(1-c)L_{II} \cdot \frac{1+k}{1-c} = (1-c)L_{II}(1+k_c)$ (15)

While the price paid by capitalists for their own consumption goods is: $cL_{II} \frac{1+k}{1-c}$ (16)

When we pass from values to prices, we may ask what happens then to surplus-value, which we know [from equation (9)] that it is equal in this new model to $L_{II} (k+c)$, and to the rate of surplus-value $k_c$, which is known to be given by equation (2), and to be equal to $\frac{k+c}{1-c}$.

Within the framework of this model, when one ceases to reason in money value in order to pass to prices, we pass also by the same token from surplus-value to profits. Let us call $\Pi_{II}$ the profits realized in the consumer goods section. The profits realized here go from $\Pi_{II} = L_I = k L_{II}$, as they were in the simple model, to:

$$\begin{align}
\Pi_{II} &= kL_{II} + cL_{II} \cdot \frac{1+k}{1-c} = L_{II} \left[ k + c \left( \frac{1+k}{1-c} \right) \right] = L_{II} \left[ \frac{k(1-c)}{1-c} + \frac{c(1+k)}{1-c} \right] = \nonumber \\
\Pi_{II} &= L_{II} \left( \frac{k+c}{1-c} \right) = L_{II} k_c \tag{17}
\end{align}$$

It must be noted that it is different from total surplus value $L_{II} (k+c)$ given by equation (9).

Finally, keeping in mind equation (2) above, if we call $\pi_{II}$ the ratio between the profits made in the consumer goods section and the price of labor in the same section, it goes from $\pi_{II} = k$ to:

$$\pi_{II} = \frac{\Pi_{II}}{W_{II}} = \frac{L_{II} \left( \frac{k+c}{1-c} \right)}{L_{II}} = \frac{k+c}{1-c} = k_c \quad (18)$$

This shows us that in the consumer goods section, when we reason in terms of prices, the new profit-to-wage ratio remains equivalent to the new rate of overall surplus value $k_c$, exactly as it was in the simple model of chapter 11.
We will need to come back to the realization of consumption goods and profits in section II, but before doing that, assuming temporarily that this realization has taken place, we will examine the realization of fixed capital goods and profits in section I, and make some calculation regarding the amount of a few variables reflecting the distribution of the whole product. These two steps will indeed make it easier to address again the question of realization in section II.

## 2.2. Realization of fixed capital goods and profits in section I

What about section I, which produces fixed capital? Before going further, let us recall that *a* remains the fraction of the total value of fixed capital accumulated in section I, and *b* the fraction accumulated in section II, with as a consequence *a* + *b* = 1. First of all, we will deal with the realization of fixed capital sold to capitalists of section II.

### 2.2.1. Fixed capital and profits realized from the sales to section II

We know that the realization of the product implies that capitalists of section II must obtain, in exchange for their profits, equal to $L_{II} \left(\frac{k+c}{1-c}\right) = L_{II}k_c$ and levied on the workers' wages, first only a fraction equal to $(1-a)L_I = bL_I$ of the money value $L_I$ of the newly produced means of production, and secondly a part of consumption goods. But because of similar conditions (which could result from competition) between capitalists of both sections, it seems logical also that capitalists of section I should obtain in addition a part of the total $cL_{II}$ money value of consumer goods accruing globally to capitalists. There is indeed no particular reason why the consumption of capitalists should be entirely reserved for those of section II, while those of section I would not have access to consumer goods! If we call $c_{II}$ the share of total consumption which will be obtained by capitalists of section II, and $c_I$ the share of total consumption which will be obtained by capitalists of section I, then we have:

$$c = c_I + c_{II} \Leftrightarrow c_I = c - c_{II} \Leftrightarrow c_{II} = c - c_I \qquad (19)$$

From this we can derive the absolute price of capitalist consumption in section I and section II, which are respectively:

$$1) \text{ For section I, } C_I = c_I L_{II} (1+k_c) \text{ or, since } L_{II} = \frac{L_I}{k}, \text{ we have also } C_I = c_I \frac{L_I}{k} (1+k_c) \quad (20)$$

$$2) \text{ For section II, } C_{II} = c_{II} L_{II} (1+k_c) = c_{II} \frac{L_I}{k} (1+k_c) \qquad (21)$$

The level of this share of consumption which goes to capitalists of section I is not predetermined, but presupposes that they must realize accordingly a profit allocated to this consumption, and which depends on the price level at which they will themselves sell first to capitalists of section II the fixed capital which the latter need, in order to implement the production of consumer goods. It is only once the price of this fixed capital is determined and
its value realized through the sale to capitalists of section II that the share of consumption of capitalists of section I in total consumption can be determined, albeit not immediately.

Firstly, these capitalists of section I must indeed deduct from their sale price the amount of wages that they have to pay to their workers, corresponding to the money value of the fixed capital which they sell to capitalists of section II, i.e. $(1-a)L_I$. The remaining amount is their gross profit, but it is not yet the profit available for the purchase of consumption goods, because they must also deduct from it the amount of their own purchase of fixed capital.

Secondly, after the sale of fixed capital of a money value of $bL_I = (1-a)L_I$, the available quantity of fixed capital has by construction a money value of $aL_I$. With the assumption that they would buy in turn, like the capitalists of section II, a fraction $b = (1-a)$ of this remaining fixed capital of a money value equal to $aL_I$, i.e. a fraction $a(1-a)L_I$, in a situation of simple reproduction, the lowest price that they could pay for this fixed capital, if the sellers did not make a profit, would be $a(1-a)L_I$. Of course this cannot be the case: therefore at first, before the purchase of this fixed capital has been made we can only know what would be the maximum amount of the difference between their gross profit and the price of the fixed capital that they buy, which is equivalent to their maximum potential consumption.

From these observations and findings we can draw an important conclusion, which is that each particular share of capitalist consumption is always a residue, depending on the price paid for the purchase of fixed capital. Firstly, the profits of capitalists of section II are a residue: their first component is the part transferred from the spending of total wages, once capitalists of section II have paid their own section II wages. The amount of this residue is decided by them when they set the price of consumption goods that they sell. Their second component comes from the sale of the remaining consumption goods to capitalists of both sections (since their own wages have already and necessarily been recovered, because it is obvious that $L_I + L_{II} > L_{II}$). Globally, however, the difference between the total price of consumption goods and total wages (equivalent to the price of workers consumption) is by construction equivalent to the price of consumption goods bought by capitalist of both sections, whose overall money value is simultaneously determined, as well therefore as the share of all capitalists in total consumption. Thus capitalist consumption as a whole can therefore be considered as a residue, its amount being simply:

$$P_{II} - W = L_{II}(1+k_c) - (L_I + L_{II}) = L_{II}(1+k_c) - L_{II}(1+k) = L_{II}(k_c - k), \text{ which is equal to:}$$

$$P_{II} - W = L_{II} \left( \frac{k+c}{1-c} - k \right) = L_{II} \left( \frac{(k+c)-k(1-c)}{1-c} \right) = L_{II} \left( \frac{c(k+1)}{1-c} \right) \Rightarrow$$

$$P_{II} - W = cL_{II} \frac{k+1}{1-c} = cL_{II}(1+k_c) \qquad (22)$$
It is easy to check that this last equation in fact gives us the exact amount of $C_c$ which we had already derived from equation (13).

Going back to profits realized by capitalists of section II, they have first to use them to buy fixed capital from capitalists of section I, otherwise simple reproduction could not be realized. And it is only the difference between these profits in section II and the price of fixed capital bought from section I which can be spent for the purchase of consumption goods. It follows that this consumption of capitalists of section II cannot but be considered itself as a residue.

Moreover the distribution of these profits of section II as a residual income, first between capitalists of both sections, and secondly among capitalists of section I, depends on the various prices charged to the buyers of fixed capital. Since it is these residual profits, when wages have been paid and fixed capital bought, that are used for buying consumption goods at a pre-determined price, it follows that capitalist consumption at each stage is always a residue.

We are now going to produce a table, given below in 2.2.2., displaying all the corresponding prices and values. In order to do that we shall first trace the various transactions of these capitalists of section I who sell fixed capital to those of section II. Capitalists producing in section I the fixed capital $I_{II}$ destined for the production of consumer goods must themselves levy on the occasion of the sale of this fixed capital to those of section II, not only the remaining fraction $aL_I$ of the value $L_I$ (that will enable them to obtain the fraction of the fixed capital $I_I$ which they need to invest), but also a fraction of the $L_{II}$ money value of consumer goods. How can they do that?

In fact this comes down to a common question of margin levying: we know that to take a margin $a$ as a fraction (or percentage) off and within a variable $y$, for instance a sale price, is mathematically equivalent to writing: $y = ay + x$, with variable $x$ being then considered as the corresponding cost price. This identity in turn implies:

$$y = ay + x \Rightarrow y(1-a) = x \Rightarrow y = \frac{x}{1-a} \Rightarrow y = x\left(1+\frac{a}{1-a}\right) \Rightarrow \frac{y}{x} = 1+\frac{a}{1-a}$$

This means simply that to levy a margin $a$ within a sale price it is necessary and sufficient to levy a markup $\frac{a}{1-a}$ over the corresponding cost price.

To apply this finding to capitalist consumption in section I, let us call $c^*$ a parameter corresponding to the ratio of consumption of all capitalists of section I which can be obtained through selling to capitalists of section II at a price $\frac{1}{1-c^*}$. The markup will thus be $\frac{c^*}{1-c^*}$.

Then we can recall that:

$$\frac{1}{1-c^*} = \frac{c^*}{1-c^*} + \frac{1-c^*}{1-c^*} \Leftrightarrow \frac{1}{1-c^*} = \frac{c^*}{1-c^*} + 1$$
Moreover it is trivial that $\frac{c^*}{\frac{1-c^*}{1-c^*}} = c^*$

All this implies that a fraction $c^*$, and thus a quantity $\frac{c^*}{1-c^*} L_I$ of the price of fixed capital $\frac{1}{1-c^*} L_I$ can be obtained when these means of production, which here have a value $bL_I = (1-a)L_I$, by definition, are sold to capitalists of section II at a price $\frac{L_I}{(1-c^*)}$ which by construction is equal to $\frac{1}{(1-a) \cdot (1-c^*)}$ times their money value $bL_I$, because this value is equal by construction to $(1-a)L_I$. Indeed we know that $\frac{1}{1-a} = \left(1+\frac{a}{1-a}\right) = \frac{1-a}{1-a} + \frac{a}{1-a}$ This means obviously that:

$$ \frac{1}{1-a} L_I = \left(1+\frac{a}{1-a}\right) L_I = \frac{1-a}{1-a} L_I + \frac{a}{1-a} L_I = L_I + \frac{a}{1-a} L_I \quad (23) $$

And multiplying by $(1-a)L_I$, we get: $(1-a)L_I \frac{1}{1-a} = (1-a)L_I \left(1+\frac{a}{1-a}\right) = (1-a)L_I + aL_I$

Since $\frac{(1-a)L_I}{(1-a) \cdot (1-c^*)}$ is obviously equal to $\frac{L_I}{1-c^*}$, this price is also equal to:

$$ \frac{L_I}{1-c^*} = \frac{L_I c^*}{1-c^*} + \frac{L_I (1-c^*)}{1-c^*} = L_I \left( \frac{c^*}{1-c^*} + 1 \right) = L_I + L_I \frac{c^*}{1-c^*} \quad (24) $$

In passing, we can develop the general formula giving the ratio of price over value from these last equations (23) and (24), for a case where there are two margins, i.e. $a$ and $c^*$ by writing:

$$ P = L \frac{1}{(1-a)(1-c^*)} = L \left[ \left( 1 + \frac{a}{1-a} \right) \left( 1 + \frac{c^*}{1-c^*} \right) \right] = L \left( 1 + \frac{a(1-c^*)+c^*(1-a)+ac^*}{(1-a)(1-c^*)} \right), \text{ which is equal to:} $$

$$ P = L \left( 1 + \frac{a - ac^* + c^* - ac^* + ac^*}{(1-a)(1-c^*)} \right) = L \left( 1 + \frac{a(1-c^*) + c^*}{(1-a)(1-c^*)} \right) \quad (25) $$
The corresponding labor price (wages) is by definition $bL_I = (1-a)L_I$, and the gross profits levied by capitalists of section I who sell to those of section II are therefore, by difference between the price of fixed capital sold to section II, i.e. $\frac{L_I}{(1-c^*)}$, and wages $(1-a)L_I$:

$$ \frac{L_I}{1-c^*} - (1-a)L_I = L_I \left[ 1 + \frac{c^*}{1-c^*} \right] - (1-a)L_I = L_I - (1-a)L_I + L_I \frac{c^*}{1-c^*} = aL_I + L_I \frac{c^*}{1-c^*} = L_I \left( a + \frac{c^*}{1-c^*} \right) \quad (26) $$

It is easy to verify that the sum of these last two terms, i.e. the price of labor $(1-a)L_I$ and the gross profits, i.e. $L_I \left( a + \frac{c^*}{1-c^*} \right)$, is indeed equal to the price of the product, i.e. the fixed capital $I_{II}$ of a money value $(1-a)L_I$ sold by capitalists of section I to those of section II:

$$ (1-a)L_I + L_I \left( a + \frac{c^*}{1-c^*} \right) = L_I + \frac{L_I c^*}{1-c^*} = L_I \left( 1 + \frac{c^*}{1-c^*} \right) = \frac{L_I}{1-c^*} \quad (27) $$

From this last equation we can derive the amount of profits of capitalists of section I who sell to those of section II. Since they come as a deduction of the profits available for consumption in section II, their amount is in fact equivalent to the overall consumption of all capitalists of section I. This allows us first to calculate the value of parameter $c_I$, i.e. the share of capitalists of section I in total consumption. This can be done in terms of $c^*$, which is the gross margin rate, and as such a given parameter, a ratio whose value is decided by capitalists of section I selling fixed capital to those of section II. In terms of prices $c_I$ is such that:

$$ L_I \frac{c^*}{1-c^*} = c_I L_{II} (1+k_c) = c_I \frac{L_I}{k} (1+k_c) \Rightarrow $$

$$ \frac{c^*}{1-c^*} = \frac{c_I}{k} (1+k_c) \Rightarrow c^* = (1-c^*) c_I \frac{1+k}{k(1-c)} $$

$$ \text{With } \gamma = \frac{c^*}{1-c^*}, \text{ we get from this last equation: } c_I = \frac{c^*}{1-c^*} \cdot \frac{k(1-c)}{(1+k)} = \gamma^* \frac{k(1-c)}{1+k} \quad (28) $$

$$ \text{This gives us reciprocally (with all calculations made): } c^* = \frac{c_I(1+k)}{k(1-c) + c_I(1+k)} \quad (29) $$

It is important to understand that $c_I$ and $c^*$, although they are both related to the consumption of all capitalists from section I, correspond to two different concepts and are thus two different variables. Indeed $c_I$ is the share of this consumption within total consumption C,
whereas c* is the gross profit margin (a percentage). Multiplied by a selling price $\frac{1}{1-c*}$ it gives $\frac{c*}{1-c*}$ which is in absolute value the fraction of the total gross profit margin intended for consumption for all capitalists from section I, but as a ratio is a markup, i.e. the ratio of this margin over the money value of capital sold to section II (on this see above, p. 233).

Therefore, if numerators of both fractions are the same, denominators are different: it is total consumption C for cI, and the money value of fixed capital III sold to capitalists of section II for c*.

These profits or this gross profit margin as an absolute amount are indeed given as an expression of Ll, i.e. the whole money value of all fixed capital produced in section I, by equation (26). But they can be expressed in terms of their money value going to section II, which is (1-a)LI, by simply multiplying both the numerator and denominator of equation (26) by(1-a). Then the equation giving the amount of profits becomes:

$$L_I \left( a + \frac{c^*}{1-c^*} \right) = (1-a) L_I \left( \frac{a}{1-a} + \frac{c^*}{(1-a)(1-c^*)} \right) = (1-a) L_I \left( \frac{a(1-c^*)+c^*}{(1-a)(1-c^*)} \right) \quad (30)$$

This equation allows us to check that equation (25) above giving the price as a function of value is perfectly correct. Indeed, remembering that the general formula giving the sale price of fixed capital sold to section II is $\frac{1}{(1-a)(1-c*)}$ times its money value, we have therefore:

$$(1-a)L_I + (1-a)L_I \left( \frac{a(1-c^*)+c^*}{(1-a)(1-c^*)} \right) = (1-a)L_I \left( 1 + \frac{a(1-c^*)+c^*}{(1-a)(1-c^*)} \right) \quad (31)$$

From this last equation (31) we can deduct that $a(1-c^*)+c^*$ is the ratio of profits over the sale price, or the total gross profit margin (used not only for consumption but also for investment) in percentage.

A last amount which can easily be calculated is that of the profits remaining to these capitalists of section I who sell fixed capital to those of section II, once they have themselves bought the fixed capital that they need from other capitalists in section I. It is indeed only the amount of these residual profits which is ultimately available for consumption. Their amount is thus the difference between their overall profits, as it had been determined by equation (26), and the price of the fixed capital that they must themselves buy, from the other capitalist of section I. It can be written:

$$\text{Residual profits} = L_I \left( a + \frac{c^*}{1-c^*} \right) - a \frac{L_I}{1-c^*} = L_I \left[ \frac{a(1-c^*)+c^*-a}{1-c^*} \right] = L_I \left( \frac{c^*(1-a)}{1-c^*} \right)$$
This amount, which corresponds to the profits available for consumption for the capitalists of section I selling fixed capital to section II, gives us therefore their consumption, which is equivalent to these profits – since we are in a system of simple reproduction, and is:

$$ (1-a)L_t \frac{c^*}{1-c^*} \qquad (32) $$

By difference with the overall consumption of capitalists of section I, i.e. $L_t \frac{c^*}{(1-c^*)}$, the consumption of capitalists of section I selling to the same section must be $aL_t \frac{c^*}{1-c^*}$. (33)

### 2.2.2. Fixed capital and profits realized from the sales within section I

Here we maintain the assumption already made within the framework of the simple model (in which parameter $u$ is equal to $b$, i.e. to $1-a$). This assumption corresponds to the second case examined in the previous chapter, according to which capitalists in section I always purchase a fraction $b = (1-a)$ of the fixed capital $(1-b) = a$, still available after the previous purchase already made of a fraction $b$.

Capitalists of section I who have sold to those of section II the fixed capital indispensable for the production of consumer goods therefore buy their own fixed capital (a fraction of $I_I$) at a price equal to $(1-b) \frac{(1-a)L_I}{(1-a)(1-c^*)} = a \frac{L_I}{1-c^*}$, and so on, and so forth, until all of the fixed capital produced for section I has been realized, knowing that all the profits realized during the successive sales of fixed capital will have been used in subsequent corresponding purchases.

To verify this, it is sufficient to draw again a table such as those in the preceding chapter, providing, for each successive purchase of fixed capital within section I, the price, the realized money value (wages) and the levied value (profits). To do that we must now consider that, instead of fixed capital being sold like in the simplified model at a price equal to $\frac{1}{1-a}$ times its money value, it is now sold at a price corresponding to $\frac{1}{1-a} \cdot \frac{1}{1-c^*}$ times its money value, including now a share of profits left available for consumption. All these data are collated in table 7 below.

For each successive level, the realized profits are also obtained, symmetrically to the calculation of equation (26) above, by difference between the price and the realized money value (which corresponds to wages paid). For the second row of the following table, we get:

$$ a \frac{L_I}{(1-c^*)} - a(1-a) L_I = aL_I \left( \frac{1}{1-c^*} - (1-a) \right) = a \left( \frac{c^*}{1-c^*} + 1 - (1-a) \right) = aL_I \left( a + \frac{c^*}{1-c^*} \right) \quad (34) $$
Similarly, profits available for consumption are obtained at each stage by difference between the overall realized profits and the price of the purchase of fixed capital, which is:

$$aL_1 \left( a + \frac{c^*}{1-c^*} \right) - a^2 \frac{L_1}{1-c^*} = aL_1 \left( \frac{a-ac^*+c^*-a}{1-c^*} \right) = aL_1 \frac{(1-a)c^*}{1-c^*} = a(1-a)L_1 \frac{c^*}{1-c^*} \quad (35)$$

The sequence of these successive purchases appears in table 12.1, where a column has been added to show the amount of profits available for consumption.

<table><caption>Table 12.1. Prices, money values and profits in section I, with capitalist consumption</caption><thead><tr><th>Realized money values = labor price = wages (with w=1)</th><th>Price of the product</th><th>Levied values = realized profits</th><th>Profits available for consumption</th></tr></thead><tbody><tr><td>(1 - a) L<sub>1</sub> = b L<sub>1</sub></td><td>(1 - a)L<sub>1</sub> / (1 - a) · (1 - c*) / (1 - c*) = L<sub>1</sub> / (1 - c*)</td><td>L<sub>1</sub> (a + c*) / (1 - c*)</td><td>(1 - a)L<sub>1</sub> c* / (1 - c*)</td></tr><tr><td>a(1 - a) L<sub>1</sub> = ab L<sub>1</sub></td><td>a L<sub>1</sub> / (1 - c*)</td><td>a L<sub>1</sub> (a + c*) / (1 - c*)</td><td>a(1 - a)L<sub>1</sub> c* / (1 - c*)</td></tr><tr><td>a<sup>2</sup>(1 - a) L<sub>1</sub> = a<sup>2</sup>b L<sub>1</sub></td><td>a<sup>2</sup> L<sub>1</sub> / (1 - c*)</td><td>a<sup>2</sup>L<sub>1</sub> (a + c*) / (1 - c*)</td><td>a<sup>2</sup>(1 - a)L<sub>1</sub> c* / (1 - c*)</td></tr><tr><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>a<sup>n</sup>(1 - a) L<sub>1</sub> = a<sup>n</sup>b L<sub>1</sub></td><td>a<sup>n</sup> L<sub>1</sub> / (1 - c*)</td><td>a<sup>n</sup>L<sub>1</sub> (a + c*) / (1 - c*)</td><td>a<sup>n</sup>(1 - a)L<sub>1</sub> c* / (1 - c*)</td></tr></tbody></table>

We recall that all quantities appearing in this table are prices, since we are in the framework of the monetary realization of the product, which is sold against money, and we have already demonstrated why these prices are pure scalars. It is easy to see that the sum of profits available for consumption is:

$$a(1-a)\left[1+a+a^2+\dots+a^n\right]L_1 \frac{c^*}{1-c^*} = aL_1 \frac{c^*}{1-c^*}, \text{ the exact amount given by equation (33).}$$
### 3. The values of a few variables reflecting the distribution of the product

We can deduct from the last table above the values taken by a certain number of characteristic quantities of the model with capitalist consumption, to begin with the price of the total product itself, a product which is the very object of distribution.

#### 3.1. The price of total product

It is important to determine the price of total product Y since it corresponds to the sum of the prices of all the final goods that are distributed. To determine this price Y we just have to add the price of consumption goods $Y_{II}$ and the price of fixed capital goods $Y_I$.

1) The price of all consumption goods $Y_{II}$ is already known from equation (12). It is:

$$Y_{II} = P_{II} = L_{II} \cdot \frac{1+k}{1-c} = L_{II} (1+k_c)$$

2) The price of fixed capital goods is obtained as follows :

The sum of the prices of fixed capital goods, which corresponds to the price of the total product of section I, i.e. $Y_I$ or $I$, is obtained by summation of the second column of table 7 above, namely:

$$Y_I = I = \frac{L_I}{1-c^*} (1+a+a^2+...+a^n) = \frac{L_I}{1-c^*} \cdot \left(\frac{1}{1-a}\right) = \frac{L_I}{(1-a)(1-c^*)} \quad \text{when } n \to \infty \quad (36)$$

The price of fixed capital $I_{II}$ for section II producing consumer goods is also:

$$I_{II} = \frac{(1-a)L_I}{(1-a)(1-c^*)} = \frac{L_I}{(1-c^*)}, \text{ which gives us, knowing that } \frac{c^*}{1-c^*} = \gamma^*$$

$$I_{II} = \frac{L_I}{(1-c^*)} = L_I \left( \frac{c^*}{1-c^*} + 1 \right) = L_I (1 + \gamma^*) \quad (37)$$

As for the price of fixed capital $I_I$ for section I, it is thus:

$$I_I = \frac{aL_I}{(1-a)(1-c^*)} = \frac{a}{1-a} L_I \frac{1}{1-c^*}, \text{ which gives us, setting } \frac{a}{1-a} = \alpha$$

$$I_I = \alpha L_I (1 + \gamma^*) = L_I (\alpha + \alpha \gamma^*) \quad (38)$$

We verify that the sum of these last two prices is equal to the price of the total fixed capital produced by section I, as provided by equation (36) above:
$$I = Y_I = I_{II} + I_I = \frac{L_I}{(1-c^*)} + \frac{aL_I}{(1-a)(1-c^*)} = \frac{(1-a)L_I + aL_I}{(1-a)(1-c^*)} = \frac{L_I}{(1-a)(1-c^*)} \quad (39)$$

$$\text{So that: } I = Y_I = L_I \left( \frac{1}{1-a} \cdot \frac{1}{1-c^*} \right) = L_I \left[ (1+\alpha)(1+\gamma^*) \right] = L_I (1+\alpha+\gamma^*+\alpha\gamma^*) \quad (40)$$

$$\text{We can check that: } Y_I = I = I_{II} + I_I = L_I(\alpha + \alpha\gamma^*) + L_I(1+\gamma^*) = L_I(1+\alpha+\gamma^*+\alpha\gamma^*) \quad (41)$$

It is a somewhat complicated expression, but it gives us the price of all fixed capital in terms of its money value $L_I$, and it follows that $(\alpha + \gamma^* + \alpha\gamma^*)$ is in fact the expression of the gross profit margin over this money value or cost price $L_I$.

$$\text{We will therefore name this ratio } \pi_I = \alpha + \gamma^* + \alpha\gamma^* \quad (42)$$

As for the price of total product $Y$, it is therefore:

$$Y = Y_I + Y_{II} = \frac{L_I}{(1-a)(1-c^*)} + L_{II} \cdot \frac{1+k}{1-c}, \text{ which gives us, setting } \frac{c}{1-c} = \gamma \Leftrightarrow \frac{1}{1-c} = 1+\gamma$$

$$Y = L_I(1+\alpha+\gamma^*+\alpha\gamma^*) + L_{II}(1+k)(1+\gamma) = kL_{II}(1+\alpha+\gamma^*+\alpha\gamma^*) + L_{II}(1+k)(1+\gamma) =$$

$$Y = L_{II}(1+\gamma) + kL_{II}(2+\alpha+\gamma+\gamma^*+\alpha\gamma^*) = L_{II}\left[(1+\gamma)+k(2+\alpha+\gamma+\gamma^*+\alpha\gamma^*)\right] \quad (43)$$

At this stage we can simplify further notations by naming $\psi$ an expression representing a good part of the second member of the above equation, so that $\psi = 1+\alpha+\gamma+\gamma^*+\alpha\gamma^*$. Replacing this value by $\psi$ in equation (43) we get:

$$Y = L_{II}\left[(1+\gamma)+k(1+\psi)\right] \quad (44)$$

## 3.2. The share of wages and profits in the product

The shares of wages and profits are indeed very important parameters, since we are here at the very heart of distribution, insofar as it is a fundamental aspect of any economic system. To determine these shares we must obviously start by displaying the amount of wages and profits, knowing that the former has already been determined.

### 3.2.1. The amount and share of wages

#### 1) The amount of wages

We set $W = W_I + W_{II}$, with $W_I$ = wages in section I, and $W_{II}$ = wages in section II

This amount is already known, since it is $W = \bar{w}(L_I + L_{II})$, which means that $W = L_I + L_{II}$ with $\bar{w} = 1$. Thus wages are $W_I = L_I$ in section I and $W_{II} = L_{II}$ and in section II.
In passing it seems interesting to verify that in section I the sum of the money values which are realized on the occasion of the sales of fixed capital corresponds to the amount of wages $W_I$ paid in this section. This sum of realized money values in section I appears in the first column of table 7, and is equal to:

$$ 
\begin{aligned}
\Sigma \text{ realized values} &= bL_I + abL_I + a^2bL_I + \dots + a^n bL_I \\
&= b \cdot L_I (1 + a + a^2 + \dots + a^n) = bL_I \frac{1}{1-a} \quad \text{when } n \to \infty
\end{aligned}
$$

Since we know that by definition $b=1-a$, then $bL_I \frac{1}{1-a} = \frac{(1-a)L_I}{1-a} = L_I$

We have thus checked that the sum of realized money values in section I is indeed $L_I$, which is the labor price $W_I$ with $\bar{w}=1$, and corresponds to the money value of the whole fixed capital produced by section I.

2) The share of wages in total product

In section II this share is rather simple, since it is $\frac{W_{II}}{P_{II}} = \frac{L_{II}}{L_{II}(1+k_c)} = \frac{1}{1+k_c}$, which gives us:

$$ \frac{W_2}{P_{II}} = \frac{1}{\frac{1+k}{1-c}} = \frac{1-c}{1+k} \qquad (45) $$

In section I the expression is the following:

$$ \frac{W_I}{P_I} = \frac{L_I}{\frac{L_I}{(1-a)(1-c^*)}} = (1-a)(1-c^*), \text{ which can also be written as:} $$

$$ \frac{W_I}{P_I} = \frac{L_I}{L_I(1+\alpha+\gamma^*+\alpha\gamma^*)} = \frac{1}{(1+\alpha+\gamma^*+\alpha\gamma^*)} = \frac{1}{1+\pi_I} \qquad (46) $$

As for the share of total wages in total product, taking the price of total product from equation (43), this share is expressed as:

$$ 
\begin{aligned}
\frac{W}{Y} &= \frac{L_I + L_{II}}{L_{II}[(1+\gamma)+k(2+\alpha+\gamma+\gamma^*+\alpha\gamma^*)]} = \frac{L_{II}(1+k)}{L_{II}[(1+\gamma)+k(2+\alpha+\gamma+\gamma^*+\alpha\gamma^*)]} \\
\frac{W}{Y} &= \frac{1+k}{1+\gamma+k(2+\alpha+\gamma+\gamma^*+\alpha\gamma^*)} = \frac{1+k}{1+\gamma+k(1+\psi)} \qquad (47)
\end{aligned}
$$

It is quite a complicated variable, which will be discussed later.
### 3.2.2. The amount and share of profits

#### 1) Profits in section II

We already know from equation (17) above that $\Pi_{II}$, i.e. the amount of profits in section II, is equal to: $$ \Pi_{II} = L_{II} \left( \frac{k+c}{1-c} \right) = L_{II} k_c \quad (48) $$

#### 2) Profits in section I

From the data in the third column of table 7 above we know that the sum of levied values in section I, i.e. the total profits realized in this section, which we name $\Pi_I$, is equal to:

$$ \Pi_I = L_I \left( a + \frac{c^*}{1-c^*} \right) \left( 1 + a + a^2 + \dots + a^n \right), \text{ which gives us, when } n \to \infty, \text{ with } \frac{c^*}{1-c^*} = \gamma^*: $$

$$ \Pi_I = L_I \left( a + \frac{c^*}{1-c^*} \right) \left( \frac{1}{1-a} \right) = L_I \left( \frac{a}{1-a} + \frac{c^*}{(1-a)(1-c^*)} \right) = L_I (\alpha + \gamma^* + \alpha\gamma^*) = L_I \pi_I \quad (49) $$

These profits are used for a part to purchase fixed capital, and for the balance to purchase consumption goods. From the data in the fourth column of the same table, we know that the sum of profits available for consumption in section I, that we name $C_I$ is equal to:

$$ \begin{aligned} C_I &= \left[ (1-a)L_I \frac{c^*}{1-c^*} \right] (1+a+a^2+\dots+a^n) = \\ C_I &= \left[ (1-a)L_I \frac{c^*}{1-c^*} \right] \left( \frac{1}{1-a} \right) = L_I \frac{c^*}{1-c^*} = L_I \gamma^* \end{aligned} \quad (50) $$

Since we know that $\frac{L_I}{1-c^*} = L_I \frac{c^*}{1-c^*} + L_I$ and that $\frac{L_I \frac{c^*}{1-c^*}}{L_I \frac{1}{1-c^*}} = c^*$, by dividing each term of the equality by $\frac{1}{1-c^*}$, we obtain the trivial identities:

$$ L_I = L_I c^* + L_I (1-c^*), \text{ or } c^* L_I = L_I - (1-c^*) L_I, \text{ or } \frac{c^*}{1-c^*} L_I = \frac{L_I}{1-c^*} - L_I $$

Conversely, it can easily be verified that the difference between $\Pi_I$, the sum of profits in section I, on the one hand, and the consumption of capitalists of this section, i.e. $C_I$, on the other hand, is equal to $I_I$, i.e. the sum of prices of fixed capital used in section I and bought by capitalists in this section. Indeed we have:

$$ \Pi_I - C_I = L_I (\alpha + \gamma^* + \alpha\gamma^*) - L_I \gamma^* = L_I (\alpha + \alpha\gamma^*) $$

And we know from equation (38) above that we have indeed $I_I = L_I (\alpha + \alpha\gamma^*)$ (51)
It is thus verified that $I_I$ is equal to $\Pi_I - C_I$, or that $\Pi_I = I_I + C_I$

3) The total amount of profits for both sections is therefore:

$$ \Pi = L_I (\alpha + \gamma^* + \alpha\gamma^*) + L_{II} k_c = k L_{II} (\alpha + \gamma^* + \alpha\gamma^*) + L_{II} k_c = L_{II} [k(\alpha + \gamma^* + \alpha\gamma^*) + k_c] \quad (52) $$

This can be written more simply as $\Pi = L_{II} (k\pi_I + k_c)$

4) The share of profits in the total product can now be calculated, as:

$$ \frac{\Pi}{Y} = \frac{L_{II} [k(\alpha + \gamma^* + \alpha\gamma^*) + k_c]}{L_{II} [(1+\gamma) + k(2+\alpha + \gamma + \gamma^* + \alpha\gamma^*)]} = \frac{k(\alpha + \gamma^* + \alpha\gamma^*) + k_c}{(1+\gamma) + k(2+\alpha + \gamma + \gamma^* + \alpha\gamma^*)} $$

Knowing that: $k_c = \frac{k+c}{1-c} = k \frac{1}{1-c} + \frac{c}{1-c} = k(1+\gamma) + \gamma$, this ratio can be somewhat simplified, to eliminate $k_c$ from the numerator, and it then becomes:

$$ \frac{\Pi}{Y} = \frac{\gamma + k(1 + \alpha + \gamma + \gamma^* + \alpha\gamma^*)}{1 + \gamma + k(2 + \alpha + \gamma + \gamma^* + \alpha\gamma^*)} $$

This last expression is quite complicated, but it can also be written more simply as:

$$ \frac{\Pi}{Y} = \frac{\gamma + k\psi}{1 + \gamma + k(1 + \psi)} \quad (53) $$

### 3.2.3. The profits/wages ratio

It is already known that $\pi_{II}$, the ratio of profits to wages in section II, is equal to the rate of surplus-value $k_c$. We can also calculate the ratio of profits over wages in section I, which we call $\pi_I$.

$$ \pi_I = \frac{\Pi_I}{W_I} = \frac{L_I (\alpha + \gamma^* + \alpha\gamma^*)}{L_I} = \alpha + \gamma^* + \alpha\gamma^* \quad (54) $$

We can check that it is exactly the same expression as that which was given by equation (42).

Using the same notations as above, we can now express more simply the ratio of total profits over total wages, which we called $\pi$, and which can be written:

$$ \pi = \frac{\Pi}{W} = \frac{L_I (\alpha + \gamma^* + \alpha\gamma^*) + L_{II} k_c}{L_I + L_{II}}, \text{ which in turn can be expressed as:} $$

$$ \pi = \frac{k L_{II} (\alpha + \gamma^* + \alpha\gamma^*) + L_{II} k_c}{L_{II} (1+k)} = \frac{k(\alpha + \gamma^* + \alpha\gamma^*) + k_c}{1+k} \quad (55) $$

By replacing $k_c$ by its expression as a function of $k$, i.e. $k_c = k(1+\gamma) + \gamma$, we get:
$$ \pi = \frac{k(\alpha + \gamma^* + \alpha\gamma^*) + k(1+\gamma) + \gamma}{1+k} = \frac{k(1+\alpha + \gamma + \gamma^* + \alpha\gamma^*) + \gamma}{1+k} = \frac{k\psi + \gamma}{1+k} \quad (56) $$

### 3.2.4. The share of capitalists from section I in total consumption

At this stage it is now possible to define the $c_I$ share which capitalists of section I get out of total consumption. By calculating the ratio of capitalist consumption in section I, i.e. $C_I$, given by equation (50), over the total price of consumer goods, i.e. $P_{IIc} = L_{II} \cdot \frac{1+k}{1-c}$, we obtain:

$$ c_I = \frac{L_I \frac{c^*}{1-c^*}}{L_{II} \frac{1+k}{1-c}} = \frac{L_I}{L_{II}} \left[ \frac{c^*(1-c)}{(1-c^*)(1+k)} \right] = k \left[ \frac{c^*(1-c)}{(1-c^*)(1+k)} \right] \quad (57) $$

We can check that this value of $c_I$ is strictly equivalent to the value that we could have calculated on a different basis, from equation (28) above, which is indeed the case, because we had found:

$$ c_I = \frac{c^* \cdot k(1-c)}{1-c^* \cdot (1+k)} \quad (58) $$

Since we know that $\frac{1-c}{1+k} = \frac{1}{1+k_c}$, this expression can also be written:

$$ c_I = \frac{c^* \cdot k}{1-c^* \cdot (1+k_c)} \quad (59) $$

Conversely we can derive from this last equation a new way of calculating $c^*$. Indeed from equation (59) we get:

$$ \frac{c^*}{1-c^*} = \frac{c_I(1+k_c)}{k}, \quad \text{which implies:} $$

$$ c^*k + c^*c_I(1+k_c) = c_I(1+k_c), \quad \text{from which we finally get:} $$

$$ c^* = \frac{c_I(1+k_c)}{k+c_I(1+k_c)} \quad (60) $$

The consumption of capitalists of section I cannot exceed a maximum constituted by the total consumption of capitalists, whose amount is determined by a fraction of the price of consumption goods fixed by capitalists of section II. We recall that this price is

$$ P_{II} = \frac{L_I + L_{II}}{1-c} = \frac{L_{II}(1+k)}{1-c} = L_{II}(1+k_c), \quad \text{and that the total consumption of capitalists } C_c \text{ is} $$

thus by definition determined by the total share of capitalists in total consumption, called c, multiplied by this price:
$$ C_c = c \frac{L_{II}(1+k)}{1-c}, \text{ or also } C_c = L_{II}\gamma(1+k) \quad (61) $$

The maximum capitalist consumption of Section I, i.e. the maximum of $C_I$, whose amount is given by equation (50), as $C_I = L_I \frac{c^*}{1-c^*}$ is therefore reached when $c_I = c$, which implies that $c_{II} = 0$, and in this case we obtain from equation (59) the following equality:

$$ c_I = k \frac{c^*(1-c)}{(1-c^*)(1+k)} = c $$

This implies that: $\frac{c}{1-c} = \frac{k}{1+k} \cdot \frac{c^*}{1-c^*}$, which can also be written:

$$ \frac{c^*}{1-c^*} = \frac{c}{1-c} \cdot \frac{1+k}{k} $$

With the notations adopted, a maximum for $c_I$ is therefore reached when we have:

$$ \gamma^* = \gamma \frac{1+k}{k} \quad (62) $$

This means that capitalists of section I could theoretically (even though this should happen only in the theoretical world) appropriate the totality of consumer goods going to capitalists, for any level of $c$, provided that the price of fixed capital sold to capitalists of section II, i.e.

$\frac{L_I}{1-c^*}$ is set at such a level of $c^*$ that this equality given by equation (62) is verified. In this case all of the profits of capitalists of section II would be used only for the purchase of fixed capital from section I and there would be no income left to them for buying consumption goods.

### 3.3. A quick consistency test

In order to ensure that the formulas in the last three sub-sections are correct, a quick check can be performed. It can for instance be verified that in section I the price of fixed capital, provided by equation (28), i.e. the price of the total product of this section, corresponding to the income of this section when all this fixed capital is realized, is equal to the sum of wages $L_I$ and profits $\Pi_I$ - given by equation (49), distributed in this section. It is indeed the case, since as can be seen in the second equation below, by performing the calculation of the numerator, the different terms of this numerator compensate and are eliminated, with the exception of $L_I$, the only remaining term.

$$ Y_I = \frac{L_I}{(1-a)(1-c^*)} = L_I(1+\alpha+\gamma^*+\alpha\gamma^*) = I_I + I_{II} \quad (63) $$
$$W_I + \Pi_I = L_I + L_I \left( \frac{a}{1-a} + \frac{c^*}{(1-a)(1-c^*)} \right) = \frac{L_I(1-a)(1-c^*) + aL_I(1-c^*) + c^*L_I}{(1-a)(1-c^*)}$$

Although this formula seems at first glance quite complicated, all the figures but one indeed cancel one another, to give in fact a much simpler formula:

$$W_I + \Pi_I = \frac{L_I}{(1-a)(1-c^*)} = L_I(1+\alpha+\gamma^*+\alpha\gamma^*) = L_I(1+\pi_I) \quad (64)$$

Finally and similarly, it can also be verified that the sum of prices of fixed capital bought by section I - given by equation (38), plus the consumption of capitalists of this same section – taken from equation (50), plus also the wages paid to its workers, is equal to the price of the whole fixed capital produced and realized by section I. This sum is:

$$Y_I = I_I + C_I + W_I = \frac{aL_I}{(1-a)(1-c^*)} + L_I \frac{c^*}{1-c^*} + L_I \quad (65)$$

Knowing that $\frac{c^*}{1-c^*} = \frac{1}{1-c^*} - 1$, this expression becomes:

$$Y_I = L_I \left[ \frac{a}{(1-a)(1-c^*)} + \left( \frac{1}{1-c^*} - 1 \right) + 1 \right]$$

$$Y_I = L_I \left( \frac{a}{(1-a)(1-c^*)} + \frac{1-a}{(1-a)(1-c^*)} \right) = \frac{L_I}{(1-a)(1-c^*)} = I_I + I_{II} \quad (66)$$

Having reached this point, all these mathematical formulas appear to be quite abstract and rather complicated, not to say somewhat esoteric. This is the reason why it seems both necessary and useful to provide a numerical example, which will be closer to the real world, and will be exposed in the following chapter. Before going to this next chapter we need nevertheless to go back to the problem of realization of products and profits in this model.

# 4. Again on the realization of the product and profits

## 4.1. A little problem of logics

At first sight, everything had seemed to go well when we touched upon the realization of the product (made of consumption goods) and profits of section II. But the attentive reader will probably have noticed that there had been in our reasoning a hidden assumption, which is that all of the consumption goods are sold, meaning not only those sold to workers, necessarily at a price equivalent to the total amount of their wages, i.e. $W_I + W_{II} = L_I + L_{II} = L_{II}(1+k)$, but also the remaining part of these consumption goods, i.e. the part sold to capitalists of both sections in exchange for an equivalent amount of their monetary profits. The price of these remaining consumption goods is necessarily equal to:
$$L_{II}(1+k_c) - L_{II}(1+k) = L_{II}k_c - L_{II}k = L_{II}\left(\frac{k+c}{1-c}-k\right) = L_{II}c\frac{1+k}{1-c} = cL_{II}(1+k_c) \quad (67)$$

It must be noted in passing that we find the same amount for capitalist consumption as that which had been calculated in another way through equations (12) to (16) above.

To buy these goods capitalists of section II could hypothetically use all the profits realized from the sale of consumption goods at a price $L_I + L_{II}$ to all workers of both sections, whereas they have paid their own workers an amount of wages equal to $W_{II} = L_{II}$: the profits made from this sale are therefore $L_I$. But in general there is no reason why these remaining consumption goods should be sold at such a price. It is only in a particular case, if the profits realized in section II from the sale to workers only are exactly equal to the price of these remaining consumption goods, that this could happen. This situation would imply that:

$$L_I = cL_{II} \frac{1+k}{1-c} \Rightarrow kL_{II} = cL_{II} \frac{1+k}{1-c} \Rightarrow k(1-c) = c(1+k) \quad (68)$$

The last equation implies that such a situation can only occur if we have either:

$$c = \frac{k}{1+2k} \quad (\text{if } k \text{ is given}), \quad \text{or} \quad k = \frac{c}{1-2c} \quad (c \text{ being given}) \quad (69)$$

There is no reason for these parameters $c$ and $k$ to be linked in such a way, but even if it were the case this would not resolve our problem of realization, because it would imply that:

- Capitalists of section I could not buy any consumption goods (all of them having already been bought by capitalists of section II);

- There would be no profits left to capitalists in section II for them to buy fixed capital from section I.

Outside of this purely hypothetical and as such quite strange situation, the problem of realization of the remaining part of consumption goods produced in section II, on top of those that can be bought from wages and from a part of profits realized in this section (the part which is not spent on fixed capital), thus remains to be solved. We must therefore find a way to solve it, which implies to make first some general remarks on money and exchanges.

## 4.2. Some remarks on money and exchanges

We have shown that money is not solely a means of measuring values and more precisely prices. We know that it is also a store of values and we have highlighted its role as a medium of exchanges. This explains why, even though wages are paid at given intervals in time, they are spent daily, i.e. on a continuous basis, which means that like production, expenditures are made in a continuous process. It is so because at the beginning of any period there is always an existing stock of money, recorded in banks' balance sheets. Moreover money, when it is
spent by a buyer and gained by a seller, does not lose its purchasing power. It only loses it when it disappears, i.e. when loans are repaid to banks which created it.

Bearing that in mind, we can infer that even though the mechanism that we described for the realization of the product implied that exchanges be logically successive, the fact that these exchanges in the real world take place over time on a continuous basis makes it possible to consider them as simultaneous, or quasi-simultaneous. As far as profits are concerned, the fact that they are realized at the same pace as the expenditure of wages, and thus equally on a continuous basis, also implies that they can also be spent on a continuous basis.

All these remarks bring us to the recognition that the mechanism highlighted in section 4 of chapter 10, in order to explain the formation of profits in section I, a mechanism first explained by Keynes who named it the widow's cruse, can be generalized and understood as working on a continuous basis. These remarks allow us to go back to the problem of the realization of all of the consumption goods produced in section II.

### 4.3. The realization of profits and product

Let us suppose first that money is borrowed by capitalists and injected in the productive system only for the payment of wages, i.e. for an amount $W_I + W_{II}$, that this whole amount is spent on consumption goods, on a continuous basis, and that the profits that they generate are spent in the same proportions as in the system described in this chapter, to buy both consumption goods and fixed capital goods. This also implies that the various exogenous parameters: $k, a, P_{II}$ (or $c$) and $c_I$ keep exactly the same value during this process.

This means that capitalists in section II, when they pay wages for an amount $W_{II}$, produce consumption goods for a value of $L_{II}$ and a price of $L_{II}(1+k_c)$, whereas they are able to sell to workers only an amount $W_I + W_{II}$ of these same goods, equivalent to $L_{II}(1+k)$. This implies that they can sell at first only a fraction $\frac{1+k}{1+k_c}$ of all the consumption goods available, which also corresponds to the workers' share of consumption goods. Since we know that:

$$1+k_c = 1+\frac{k+c}{1-c} = \frac{1+k}{1-c}, \text{ it follows that } \frac{1+k}{1+k_c} = \frac{1+k}{1+k} = 1-c \qquad (70)$$

Not surprisingly, the share of workers is thus $1-c$, as could be expected. What about the initial profits that capitalists can make from this sale? To be sure, the value of consumption goods sold at a price $L_I + L_{II} = L_{II}(1+k)$ is lower than $L_{II}$, because capitalists do not sell to workers all the produced consumption goods but only a fraction, the value of which is equivalent to the overall price of goods sold to workers divided by $1+k_c$ (the ratio of price over value). This fraction is therefore $\frac{L_{II}(1+k)}{1+k_c} = L_{II}(1-c)$, as just seen. However these
initial profits in section II are not the difference between this price $L_{II}(1+k)$ and this last amount, because the mere fact that capitalists in this section are able to sell their goods at this price $L_{II}(1+k)$ implies that they have already paid previously these whole wages for an amount $L_{II}$, and not $L_{II}(1-c)$.

These initial profits, which they can spend, are therefore the difference between the reduced volume of sales and the amount of wages $W_{II} = L_{II}$ which they have paid actually, and they are also lower than the full profits of section II determined at the beginning of this chapter, in subsection 2.1., from equation (17). Therefore, instead of being $k_c L_{II}$, this initial amount of profits, which for this reason we will name $\Pi_{II}^1$, must be considered as the difference between the amount sold to workers and the whole wages paid in section II:

$$ \Pi_{II}^1 = L_{II}(1+k) - L_{II} = kL_{II} \qquad (71) $$

This last amount in fact corresponds to the amount of profits in a model without capitalist consumption and represents only a percentage of $k_c L_{II}$, i.e. the amount of profits which would be obtained by the sale of the whole product of section II. This percentage is:

$$ \frac{\Pi_{II}^1}{\Pi_{II}} = \frac{kL_{II}}{k_c L_{II}} = \frac{k}{k_c} = \frac{k}{\frac{k+c}{1-c}} = \frac{k(1-c)}{k+c} \qquad (72) $$

From this equation it is quite easy to express these initial profits in terms of the final or targeted ones:

$$ \Pi_{II}^1 = \frac{k(1-c)}{k+c} \Pi_{II} = \frac{k(1-c)}{k+c} k_c L_{II} \cdot \text{It is easy to check that this is equivalent to:} $$

$$ \Pi_{II}^1 = \frac{k(1-c)}{k+c} \left[ \frac{k+c}{1-c} L_{II} \right] = kL_{II}, \text{ which is not a surprise!} $$

From this we see that the amount of money profits in section II, which by construction is equivalent to surplus-value, cannot initially be increased by the rise in the price of consumption goods from $L_{II}(1+k)$ to $L_{II}(1+k_c)$, compared to a situation without capitalist consumption, through the sale of these goods to workers, as long as the remaining part of consumption goods has not itself been realized by being sold to capitalists of both sections, who have to spend a part of their profits to this end. Even if capitalists of section II used the totality of this realized amount of profits to buy consumption goods, they would obtain a share of these goods equivalent to:
$$ \frac{\Pi_{II}^{1}}{P_{II}} = \frac{kL_{II}}{(1+k_{c})L_{II}} = \frac{k}{1+\frac{k+c}{1-c}} = \frac{k(1-c)}{1+k} \qquad (73) $$

At the same time, the share of consumption goods available after workers have bought their own share is itself equivalent to:

$$ \frac{(1+k_c)L_{II} - (1+k)L_{II}}{(1+k_c)L_{II}} = \frac{\frac{1+k}{1-c} - 1 + k}{1+k} = \frac{1+k - (1+k)(1-c)}{1+k} = 1 - (1-c) = c \qquad (74) $$

It is again not a surprise to find out that this share is c, i.e. the share of total capitalist consumption from both sections out of the total production of consumption goods.

Thus if we want to compare the share of initial profits $\Pi_{II}^1 = kL_{II}$, to capitalist consumption, i.e. $C_c = c(1+kc)L_{II} = c\frac{1+k}{1-c}L_{II}$, there is a value of parameters such that $\Pi_{II}^1$ and $C_c$ are equal. This is the case when $kL_{II} = c\frac{1+k}{1-c}L_{II}$, which implies that we should have between k and c the following relation: $\frac{k}{1+k} = \frac{c}{1-c}$

But this is indeed quite a particular case, and if we leave it to go back to the most general case, what we have in the system when the initial profits of section II materialize is an amount of monetary profits $\Pi_{II}^1 = kL_{II}$, which is lower than the targeted amount of profits, i.e. $k_cL_{II} = \frac{k+c}{1-c}L_{II}$, by a factor $\frac{k(1-c)}{k+c}$, and these initial profits are not sufficient to realize the purchase of both consumption goods and fixed capital needed by section II for the amounts which would allow for the realization of the product in both sections.

It seems therefore appropriate to adopt a realistic approach, by supposing that these initial profits are spent exactly in the same proportions which would allow for the realization of the whole product if these profits were of the targeted amount. The problem then consists in understanding what happens when all the parameters which constitute the exogenous variables remain unchanged, and when we start the spending of profits from section II with an amount of profits which is $kL_{II}$ instead of being supposed immediately to be $k_cL_{II}$, as we postulated above in section II of the present chapter.

Indeed, if the same proportion of section II profits is invested in the purchase of fixed capital goods from section I, it means that instead of investing from the beginning an amount
$I_{II} = \frac{L_I}{1-c^*}$, section II will at first only be able to invest an amount reduced in similar

$$\text{proportions } \frac{k}{k_c}, \text{ i.e. to } I_{II}^1 = \frac{kL_I}{k_c(1-c^*)} = \frac{k(1-c)L_I}{(k+c)(1-c^*)} \quad (75)$$

This in turn will affect in a similar proportion the profits of section I, which will reduce the amount of total investments of section I to:

$$I_I^1 = \frac{kaL_I}{k_c(1-a)(1-c^*)} = \frac{k(1-c)aL_I}{(k+c)(1-a)(1-c^*)}, \text{ and the total product of section I to:}$$

$$Y_{II}^1 = I_I^1 = I_I^1 + I_{II}^1 = \frac{kL_I}{k_c(1-a)(1-c^*)} = \frac{k(1-c)L_I}{(k+c)(1-a)(1-c^*)}.$$

Finally it is the amount of total capitalist consumption for both sections that in turn at this stage, will not be:

$$C_c = cL_{II}(1+k_c) = cL_{II}\frac{1+k}{1-c}, \text{ but only a } \frac{k}{k_c} \text{ proportion of it:}$$

$$C_c^1 = \frac{k}{k_c}C_c = \frac{k}{k_c}c(1+k_c)L_{II} = \frac{k(1-c)}{k+c}\left(\frac{1+k}{1-c}\right)cL_{II}, \text{ which is equivalent to:}$$

$$C_c^1 = c \frac{1+k}{k+c} kL_{II}. \quad (76)$$

To summarize our approach, we introduced capitalist consumption in the model through an increase in surplus-value and profits in section II based on an initial increase in the prices of consumption goods (compared to the situation without capitalist consumption). Then we showed that the expense of wages generates profits in section II for an amount $\Pi_{II}^1 = kL_{II}$. In turn we just showed that these profits generate a capitalist consumption $C_c^1 = c \frac{1+k}{k+c} kL_{II}$.

A key element in this demonstration is now to recognize that this capitalist consumption corresponds to pure additional profits in section II, for the simple reason that the whole amount of wages paid in this section (i.e. $W_{II} = L_{II}$) has already been deducted from the sale price of consumption goods bought by workers, in the calculation of $\Pi_{II}^1$, and can thus already be repaid to the banks from which this amount has initially been borrowed. These profits, which we name $\Pi_{II}^2$, are therefore exactly equal to this $C_c^1$ amount of consumption, and therefore we can write:
$$ \Pi_{II}^2 = C_c^1 = c \frac{1+k}{k+c} k L_{II} \qquad (77) $$

In turn these profits will generate an additional consumption by capitalists from both sections, which – with all parameters remaining unchanged, is necessarily proportional to the ratio of these new profits in section II to the total profits of this section, and such that we can write:

$$ \frac{C_c^2}{C_c} = \frac{\Pi_{II}^2}{\Pi_{II}} \Rightarrow C_c^2 = C_c \frac{\Pi_{II}^2}{\Pi_{II}} = $$

$$ C_c^2 = c(1+k_c)L_{II} \left( \frac{ck \frac{1+k}{k+c}}{k_c} \right) = c \left( \frac{1+k}{1-c} \right) \left( \frac{ck \frac{1+k}{k+c}}{1-c} \right) L_{II} = c \left( \frac{1+k}{1-c} \right) \cdot ck \left( \frac{1+k}{k+c} \cdot \frac{1-c}{k+c} \right) L_{II} = $$

$$ C_c^2 = c^2 k \frac{(1+k)^2}{(k+c)^2} L_{II} = k c^2 \left( \frac{1+k}{k+c} \right)^2 L_{II} \qquad (78) $$

In turn this consumption corresponds to pure profits $\Pi_{II}^3 = C_c^2$, which can themselves be spent again, thus generating new consumption and new profits, and so on, and so forth, ad infinitum, so that the sum of all these profits in section II can be written:

$$ \sum_{i=1}^{n} \Pi_{II}^{i} = \Pi_{II}^{1} + \Pi_{II}^{2} + \dots + \Pi_{II}^{n} = $$

$$ \sum_{i=1}^{n} \Pi_{II}^{i} = kL_{II} \left[ 1 + \frac{c(1+k)}{k+c} + \left( \frac{c(1+k)}{k+c} \right)^2 + \dots + \left( \frac{c(1+k)}{k+c} \right)^n \right] \qquad (79) $$

We recognize the sum of a geometric series, which converges when $n \to \infty$, because its common ratio is $\frac{c(1+k)}{k+c}$, which is obviously less than 1 since c is necessarily lower than 1, with that we have always $ck < k$. Using the well-known formula for such a sum, we thus get:

$$ \Pi_{II} = \sum_{i=1}^{n} \Pi_{II}^{i} = kL_{II} \left[ \frac{1}{1 - \frac{c(1+k)}{k+c}} \right] = kL_{II} \left[ \frac{k+c}{k(1-c)} \right] = L_{II} k_c \qquad (QED) \qquad (80) $$

As for capitalist consumption, since we have $C_{II}^n = \Pi_{II}^{n+1}$ it is easy to see from equation (79) that the sum of the successive amounts of capitalist consumption is equal to:
$$ \sum_{i=1}^{n} C_{c}^{i} = \sum_{i=1}^{n} \Pi_{II}^{i} - \Pi_{II}^{1} = \Pi_{II} - kL_{II} = (k_{c} - k)L_{II} \Rightarrow $$

$$ C_c = \left( \frac{k+c}{1-c} - k \right) L_{II} = \frac{c(1+k)}{1-c} L_{II} = c(1+k_c) L_{II} \quad (\text{QED}) \quad (81) $$

We have thus demonstrated that when exogenous parameters stay unchanged, the spending of the initial profits of section II, through the successive spending of section I and section II profits, allows for the full realization of the full profits of section II (equation 79) and for the full realization of capitalist consumption. The conclusion that can be drawn from the above demonstration, is that when money injected in the productive system only corresponds to the amount of wages, it does allow for the realization of the full product in the sphere of circulation, because the expected amount of profits and capitalist consumption in section II and consequently in section I, can be realized, through the successive expense of profits.

In passing, we can also confirm that setting the price of consumption goods to $P_{II} = L_{II}(1+k_c)$ ensures to reach the targeted amount of capitalist consumption, as a share $c$ of indifferently the value or the price of consumption goods. In other words with the formulation of the rate of surplus-value as $k_c = \frac{k+c}{1-c}$ which we used so far, we get indeed:

$$ \begin{aligned} C_c &= P_{II} - (L_I + L_{II}) = L_{II} \left(1 + \frac{k+c}{1-c}\right) - L_{II} (1+k) = L_{II} \left(1 + \frac{k+c}{1-c} - (1+k)\right) = \\ &= L_{II} \left(\frac{k+c-k+kc}{1-c}\right) = L_{II} c \left(\frac{1+k}{1-c}\right) \end{aligned} $$

And since we know that $\frac{1+k}{1-c} = 1 + \frac{k+c}{1-c} = 1 + k_c$, then we have $C_c = cL_{II}(1+k_c)$

It is also immediate that the ratio of capitalist consumption $C_c$ over the total price of consumption goods is:

$$ \frac{C_c}{P_{II}} = \frac{cL_{II}(1+k_c)}{L_{II}(1+k_c)} = c $$

We have thus demonstrated once more that the formulation of the rate of surplus-value (and share of profits in section II) which we have used so far is perfectly correct, and allows for the full realization of the whole product in both sections of the productive system.

The reason for such an outcome has to do with the nature of money as a store of value. Indeed when money is created at the stage of production to pay for wages, since - as we have demonstrated earlier, money wages are the measure of the money value of the product, this money has a purchasing power which corresponds exactly to this money value. When workers
purchase consumption goods they do not get the value corresponding to what they have produced, but only a share $\frac{1}{1+k_c}$ because the price of these goods exceeds their money value by a factor $1+k_c$. And the remaining part $k_c$, which corresponds to surplus-value, has kept its purchasing power, which has been converted in monetary profits.

Consequently it is simply this same mechanism, which reduces by a factor $\frac{1}{1+k_c}$ the consumption of workers, while transferring a share of value $k_c$ to capitalists of section II, which is also at work with capitalists expenditures. We had shown in subsection 2.2. above that, assuming that all profits had been realized in section II, this mechanism ensured the realization of profits and fixed capital goods in section I producing this fixed capital. We have now shown that it also works for the realization of profits and product in section II, i.e. for the consumption of capitalists.

Indeed the fact that to reduce workers consumption (and free a part of consumption goods for capitalist consumption) the price of commodities has to be increased to the necessary extent, compared to their money value, to be sure also reduces the amount of money value – i.e. the purchasing power of money, which can be realized by capitalists in the spending of profits to buy consumption goods, but at the same time it transfers the remaining part, which keeps its purchasing power, to other capitalists! And the same price differential over value works in section I for transferring purchasing power on fixed capital among capitalists.

Indeed, as we have seen so far, any increase in the price of any goods by adding to the initial value a percentage $x$ is equivalent to levying a percentage $\frac{x}{1+x}$ out of the purchase at this increased price $1+x$ or to multiplying the initial value by $1+x$. But then this primary levy also leads to the creation of a secondary levy of the same percentage $x$ on the purchasing power of all secondary incomes derived from these primary incomes, when they are spent on the same goods. Thus these secondary incomes themselves can obtain only a $\frac{1}{1+x}$ fraction of the increased price of these goods (because $1 - \frac{x}{1+x} = \frac{1}{1+x}$). This reduction in the purchasing power of money incomes in terms of the money value which they can obtain, transfers however a fraction $\frac{x}{1+x}$ of the purchasing power. This gives us a new principle.

**Principle 23:** A levy imposed on the purchasing power of primary incomes through an increase in the price of consumption goods also constitutes a transfer of purchasing power to secondary money incomes. Spending these incomes creates a new levy on secondary incomes when they buy consumption goods as well as fixed capital, and a new transfer of purchasing power. The same process can continue as long as purchasing
power is remaining and ensures henceforth the full realization of the whole product
from both sections of the productive system.

It is therefore obvious that the only money which is really needed in the system corresponds
to money created as a counterpart for production and labor incomes, since it allows for the full
realization of the product from both section II and section I. It is so because this money first
allows for an initial realization of profits in section II, which is used to buy consumption
goods as well as fixed capital in the appropriate amounts. At the same time these expenditures
also transfer purchasing power to capitalists and among them, a process which converges up
to the full expense of this purchasing power.

Although Keynes's remarks on the widow's cruse concerned only the consumption of
entrepreneurs (as he called capitalists), we had previously generalized this mechanism to
account for the formation of profits in section I. It is therefore not surprising to see that his
remarks account for section II and capitalist consumption, and are therefore fully validated.
Indeed it is worth quoting him again: "There is one peculiarity of profits (or losses) that we
may note *in passing*, because it is one of the reasons to segregate them from income proper, as
a category apart. If entrepreneurs choose to spend a portion of their profits on consumption
[...] the effect is to increase the profit on the sale of liquid consumption goods by an amount
exactly equal to the amount of profits which have been thus expended [...] Thus, however
much of profits entrepreneurs spend on consumption, the increment of wealth belonging to
the entrepreneurs remains the same as before. Thus, profits, as a source of capital increment
for entrepreneurs, are a widow's cruse which remains un-depleted, however much of them
may be devoted to riotous living" (Keynes, 1930, p. 139).

Keynes remarks were not really supported by a full and detailed explanation, which we just
provided, expanding its range of action to the whole sphere of circulation. The mechanism is
undeniable, and it goes much further than Keynes himself imagined, because we expanded it
to investment in fixed capital within section I. This mechanism is however only true and can
only work on the basis of a few conditions which make it possible. Indeed as we have
demonstrated, in order to increase their consumption by spending their profits, if we start
from a situation without capitalist consumption, capitalists will increase their profits from
$kL_{II}$ to $k_c L_{II}$. What we have shown it that this implies to raise the rate of surplus-value from
$k$ to $k_c$, which has to be achieved through an increase in the price of consumption goods. As
for the share of profits devoted - as Keynes writes it, to "riotous living", i.e. to the
consumption of luxury goods, it needs a change in the composition of consumption goods
towards an increased production of this kind of goods.

One other thing that was not mentioned by Keynes is that this widow's cruse mechanism
cannot work indefinitely, and has a limit. To be sure, as we have demonstrated, capitalist
consumption translates into profits for an identical amount, just as Keynes had indicated: "the
effect is to increase the profit on the sale of liquid consumption goods by an amount exactly
equal to the amount of profits which have been thus expended" (ibidem).
This is true because wages have already been paid in the sector of consumption goods, but these profits cannot in turn be used by capitalists of section II only to buy other consumption goods, because a part of them must necessarily be spent on the purchase of fixed capital. Otherwise the whole system would be blocked: if capitalists of section I did not sell fixed capital goods to those of section II, they could not purchase consumption goods at the next stage of spending. This is the reason why these profits will translate into a lower amount of expenditure on consumption goods at the next stage, which will make the system gradually converge towards its built-in levels of profits and capitalist consumption. This is all the more needed that there is in fact a limit on the consumption of capitalists, which is the value of consumption goods available once workers have themselves bought their share of these goods. It must also be understood that, at the completion of the process, all monetary profits have been spent and have thus progressively disappeared, as well as money wages.

## References

Keynes, John Maynard (1930). *A Treatise on Money*. Last edition: 1971. London Macmillan.
# Chapter 13. A numerical example of a model with capitalist consumption

In this chapter we will provide an example, and will first start by recalling what are the exogenous variables that appear in it. This will allow us to then show how it is possible to determine from them a certain number of endogenous variables. This will make it possible finally to calculate the amounts of all the variables relating to the distribution of the product.

## 1. Exogenous variables

The first exogenous variable is the value of the product and its distribution between the two sections, which are essentially technological data:

- The product social value is $Y = L = 150$ (in time units), which is distributed between a value $Y_I = L_I = 50$ for the first section producing fixed capital, and a value $Y_{II} = L_{II} = 100$ for the second section producing consumption goods.

- Taking also the average nominal wage per unit of time as a unit, equal to 1, i.e. $\bar{w}=1$, then the money value of the product is $W = \bar{w}L = L=150$, with $W_I = \bar{w}L_I = L_I = 50$ and $W_{II} = \bar{w}L_{II} = L_{II} = 100$.

This implies that the ratio of the product social value in both sections $k = \frac{L_I}{L_{II}}$ is also an exogenous variable. It is essentially a technical parameter, although it reflects the distribution of value between the two sections of production, and from its definition in the first simplified model may be considered as the primary rate of surplus-value.

- In this numerical example it is: $k = \frac{L_I}{L_{II}} = \frac{50}{100} = 0.5$

The second variable, which is also a technological datum, is the share in value of fixed capital going to section I, for the production of fixed capital, over the total quantity of fixed capital. It is named $a$, and since it is the ratio of investment in section I to total investment, we have, for this numerical example:

$$a = \frac{I_I}{I} = 0.4 = 40 \%$$

Since the share of total fixed capital going to section II, called $b$, is obtained by construction from its difference with $a$, the share of total fixed capital going to section I, to which it is the complement to 1, it must similarly be considered as a complementary exogenous variable. It means that in this example its value is necessarily:
$$b = 1 - a = 1 - 0.4 = 0.6 = 60 \%$$

A third variable is the price of consumption goods produced by section II, which is 200, in money units. So we have $P_{II} = 200$. But $P_{II}$ is also by definition and equivalently the amount of total consumption: $C = 200$, for a money value equal to $L_{II} = 100$.

Although that may seem difficult to grasp at this stage in view of the apparent complication of the model, the only other exogenous variable is $c_I$, the share of consumption of capitalists of section I over total consumption. Let us consider that $c_I = 0.15 = 15 \%$.

Indeed all the other variables derive from these four variables, as it will now be demonstrated.

## 2. Endogenous variables

The first endogenous variable is the consumption of workers, which is identical by construction to the overall amount of wages for both sections, since in a situation of simple reproduction all wages are spent in the purchase of consumption goods, i.e.:

$$C_W = L_I + L_{II} = W_I + W_{II} = 50 + 100 = 150$$

We can also obtain directly from this last figure the share of workers in total consumption, the second endogenous variable, which we call $c_W$, and which is:

$$c_W = \frac{C_W}{C} = \frac{150}{200} = 0.75 = 75 \% \text{ (it is also equivalent to } c_W = 1 - c = 1 - 0.25 = 0.75 \text{)}$$

Workers can therefore buy 75 % of the total price of consumption goods, and identically 75 % of their value (i.e. $L_{II} = 100$), which amount to 75. This gives us at the same time the value of labor-power, since it is equal by definition to the value of the consumption goods which can be purchased by workers. We call $V$ this third variable and thus have $V = 75$.

From the amount of workers consumption $C_W$ we have immediately by difference a fourth endogenous variable, the consumption of capitalists, which is:

$$C_c = C - C_W = 200 - 150 = 50$$

From this last amount we immediately get a fifth endogenous variable, which is the share of capitalist consumption in total consumption:

$$c = \frac{C_c}{C} = \frac{50}{200} = 0.25 = 25 \%$$

From $c$ and $c_I$ (an exogenous variable) we can directly obtain the share of consumption of capitalists of section II over total consumption. As we know it is called $c_{II}$ and is our sixth
endogenous variable, calculated by difference, because by definition $c = c_I + c_{II}$, which gives us therefore:

$$c_{II} = c - c_I = 0.25 - 0.15 = 0.10 = 10 \%$$

Now, the share of capitalist consumption c out of total consumption gives us our certainly most important endogenous variables (the seventh and eight ones), since they are nothing less than:

- First, the amount of surplus-value S, i.e. the value of fixed capital plus the value of capitalist consumption:

$$S = L_I + C_c = 50 + 25 = 75$$

- Second, the actual rate of surplus-value for this model, which we have defined as $k_c$, and which is by construction:

$$k_c = \frac{k+c}{1-c} = \frac{0.5+0.25}{1-0.25} = 1 = 100 \%$$

We can easily check that this rate of surplus-value is strictly identical to the rate which can also be calculated from two other different formulas:

- The first one based on the price of consumption goods, knowing that $P_{II} = L_{II}(1+k_c)$, which gives indeed the rate of surplus-value as:

$$k_c = \frac{P_{II} - L_{II}}{L_{II}} = \frac{200-100}{100} = 1 = 100 \%$$

- The second one based on the ratio of surplus-value over the value of labor-power, which is:

$$k_c = \frac{S}{V} = \frac{75}{75} = 1 = 100 \%$$

A last variable which must be determined is $c^*$, which as we have seen is a parameter linked to $c_I : c^*$ has been defined in the last chapter as corresponding to the share of consumption of capitalists of section I selling to capitalists of section II contained in a price $\frac{1}{1-c^*}$, this share being thus $\frac{c^*}{1-c^*}$. It must be understood that it is not the full profit margin, since a part of this profit margin is being spent for the purchase of fixed capital, but only the remaining part of this profit margin, which can be realized after fixed capital has been bought. Indeed we have seen that the sale price itself is in fact $\frac{1}{(1-a)(1-c^*)}$ times the cost price (or money value).
This remaining share is therefore available for consumption and thus spent on consumption goods.

We know from equation (58) that $c^* = \frac{c_I(1+k_c)}{k+c_I(1+k_c)}$, so that with $c_I = 0,15$ this amount is:

$$c^* = \frac{0.15(1+1)}{0.5+0.15(1+1)} = \frac{0.3}{0.8} = 0.375 \text{ or } 37.5 \%$$

It follows that the profit margin intended for consumption over the rest of the price is:

$$\frac{c^*}{1-c^*} = \frac{0.375}{1-0.375} = \frac{0.375}{0.625} = 0.6 = 60 \%$$

Now that all the main variables have been defined or calculated, we are able to see easily how the income and product are distributed among all stakeholders.

# 3. The distribution of income and product

## 3.1. In section II

In section II producing consumption goods, production has a money value of 100, equal to the wage bill in this section ($W_{II} = L_{II} = 100$) and a monetary price of 200, which gives an amount of profits of 100. In passing it is noteworthy that, as previously pointed out, the profits/wages ratio in section II, i.e. $100/100 = 1$, is exactly equal to the rate of surplus-value $k_c$ of 100 %.

These monetary profits are obtained by successive sequences of spending:

The purchase of consumption goods by workers from their wages $W_I + W_{II} = L_I + L_{II} = 150$ generates an amount of initial profit $\Pi_{II}^1 = kL_{II} = 0.5 \times 100 = 50$

These initial profits are then spent, both on consumption goods and fixed capital, by capitalists of section II, and capitalists of section I spend themselves a part of the profits which they make from this spending for the purchase of consumption goods, for an amount which from equation (75) in the previous chapter is equal to:

$$C_c^1 = c \frac{1+k}{k+c} kL_{II} = \frac{0.25(1+0.5)}{0.5+0.25} 0.5 \times 100 = \frac{0.375}{0.75} 50 = 25$$

This last amount is itself equal to $\Pi_{II}^2 = 25$

The same phenomenon is reproduced from this second “wave” of profits $\Pi_{II}^2$, to obtain an amount of consumption $C_c^2$ which from equation (77) in the previous chapter is equal to:
$$C_c^2 = kc^2 \left(\frac{1+k}{k+c}\right)^2 L_{II} = 0.5(0.25)^2 \left(\frac{1+0.5}{0.5+0.25}\right)^2 100 = 0.5(0.0625)(4)100 = 0.125 * 100 = 12.5$$

This last amount is itself equal to $\Pi_{II}^3 = 12.5$

From the last chapter, we know that this series continues and converges to obtain finally a sum of profits realized and spent in section II, which - as expected - is given by equation (78):

$$\Pi_{II} = kL_{II} \left[ \frac{k+c}{k(1-c)} \right] = k_c L_{II} = 1 * 100 = 100$$

We have shown also that these profits correspond to a total amount for capitalist consumption which is, from equation (79):

$$C_c = \frac{c(1+k)}{1-c} L_{II} = c(1+k_c) L_{II} = 0.25(1+1) * 100 = 50$$

With their total profits amounting to $\Pi_{II} = 100$, capitalists of section II must buy the quantity of fixed capital which they need, i.e. by construction $b=1-a=1-0.4=0.6=60\%$ of the value of all fixed capital available, which is 60 % of $L_I = 50$, corresponding to an amount of 30, in money value.

The price that they have to pay for this fixed capital is $(1-a)L_I \frac{1}{(1-a)(1-c*)} = \frac{L_I}{1-c*}$

$$\frac{L_I}{1-c^*} = L_I \left(1 + \frac{c^*}{1-c^*}\right) = 50 \left(1 + \frac{0.375}{1-0.375}\right) = 50 \left(1 + \frac{0.375}{0.625}\right) = 50(1+0.6) = 80$$

As a result, the remaining profits available for consumption by capitalists of section II once this fixed capital is paid for are:

$$C_{II} = L_{II}k_c - L_I \frac{1}{1-c^*} = L_{II}k_c - kL_{II} \frac{1}{1-c^*} = L_{II} \left( k_c - \frac{k}{1-c^*} \right) = 100 \left( 1 - \frac{0.5}{0.625} \right) = 100(1-0.8)$$

$$C_{II} = 100 - 80 = 20$$

This implies that $c_{II} = \frac{20}{200} = 0.1 = 10\%$ of total consumption.

## 3.2. In section I

1) Capitalists selling fixed capital to section II sell it as already seen for a price of 80, for a money value (wages) of 30, leaving them an amount of profits of 50.

Fixed capital available for section I after this sale has a money value of $aL_I = 0.4 \times 50 = 20$
In line with the assumption previously made, they buy a fraction $b = 1 - a = 0.6$ of the value of this capital, i.e. 60% of 20 = 12

From their profits of 50 we know from table 7 in chapter 12 that for the purchase of this fixed capital they pay a price of $a \frac{L_I}{1-c^*}$, which is equal to $0.4 \frac{50}{1-0.375} = \frac{20}{0.625} = 32$

The balance of profits available for their consumption is therefore: 50 - 32 = 18, which allows them to buy consumption goods for an identical price of 18, and a value of 9.

2) At the next level, capitalists selling fixed capital to the previous ones in section I sell it for a price of 32, corresponding to a money value of 12 (wages), as we just saw, leaving them with an amount of profits of 20.

Fixed capital available after this sale has a money value of 20 - 12 = 8

In line with the assumption previously made, they buy a fraction $b = 1 - a = 0.6$ of this capital, i.e. 60% of 8 = 4.8

From their profits we know from table 7 in last chapter that to buy this fixed capital they pay a price of $a^2 \frac{L_I}{1-c^*}$, which is equal to $0.16 \frac{50}{1-375} = \frac{8}{0.625} = 12.8$

The balance of profits available for consumption is therefore: 20 - 12.8 = 7.2 which allows them to buy consumption goods for an identical price of 7.2, and a value of 3.6.

3) At the next level, capitalists selling fixed capital to the previous ones in section I sell it for a price of 12.8, for a money value of 4.8 (wages), leaving them an amount of profits of 8.

Fixed capital available after this sale has a money value of 8 - 4.8 = 3.2

In line with the assumption previously made, they buy a fraction $b = 1 - a = 0.6$ of this capital, i.e. 60% of 3.2 = 1.92

From their profits we know from table 7 in last chapter that to buy this fixed capital they pay a price of $a^3 \frac{L_I}{1-c^*}$, which is equal to $0.064 \frac{50}{1-375} = \frac{3.2}{0.625} = 5.12$

The balance of profits available for consumption is therefore: 8 - 5.12 = 2.88 which allows them to buy consumption goods for an identical price of 2.88, and a value of 1.44.

4) It does not seem necessary to continue this demonstration, all the more so that we can see that after only three stages in the sales of fixed capital the total amount of these sales in money value has already reached 12 + 4.8 + 1.92, which makes a total of 18.72 out of a value of fixed capital intended for section I of 20 (50 minus 30 sold to section II). Thus 93.6 % of this amount of 20 have already been realized, after only three “rounds”!

5) For section I as a whole we will have the following figures:
i. The total money value of fixed capital produced is as we know $L_I = W_I = 50$

ii. The total price of fixed capital sold to this section and bought within it (which as we know has a value of $aL_I = 0.4 \times 50 = 20$), is given by equation (30), as:

$$I_I = \frac{aL_I}{(1-a)(1-c^*)} = \frac{0.4 \times 50}{(1-0.4)(1-0.375)} = \frac{20}{0.6 \times 0.625} = \frac{20}{0.375} = 53.33$$

iii. The total amount of profits realized in this section (including profits from the sale of fixed capital to section II) is given by equation (39) as:

$$\Pi_I = L_I \left( \frac{a}{1-a} + \frac{c^*}{(1-a)(1-c^*)} \right) = 50 \left[ \frac{0.4}{0.6} + \frac{0.375}{0.6 \times 0.625} \right] = 50(0.667 + 1) = 83.33$$

iv. The total of profits available for consumption in this section is provided for each successive stage by table 7 in the previous chapter as:

$$(1-a)L_I \frac{c^*}{1-c^*} + a(1-a)L_I \frac{c^*}{1-c^*} + a^2(1-a)L_I \frac{c^*}{1-c^*} + \dots + a^n(1-a)L_I \frac{c^*}{1-c^*} =$$

$$(1-a)L_I \frac{c^*}{1-c^*} (1+a+a^2+\dots+a^n) = (1-a)L_I \frac{c^*}{1-c^*} \left(\frac{1}{1-a}\right) = L_I \frac{c^*}{1-c^*}$$

This gives us, as the total amount of profits available for consumption in section I:

$$\Pi_I - I_I = 83.33 - 53.33 = 30$$

We can check that is exactly the same amount as that derived from the above equation:

$$L_I \frac{c^*}{1-c^*} = 50 \frac{0.375}{0.625} = 50 \times 0.6 = 30$$

We can also check that it is indeed equal to $C_I$, i.e. the consumption of capitalists of section I, whose share $c_I$ in total consumption is exogenous and given as $c_I = 0.15$. $C_I$ is indeed:

$$c_I C = 0.15 C = 15\% \times 200 = 30$$

We have thus checked that all these figures for all of these different endogenous variables correspond perfectly to all the equations that we had devised in order to calculate them from the exogenous variables. This validates these equations.
# 4. The values of a few variables reflecting the distribution of the product

## 4.1. The price of total product

The price of total product $Y$ is equal to the price of consumption goods $Y_{II} = 200$ plus the price of fixed capital goods $Y_I$, which is itself the sum of prices of fixed capital sold to section II, i.e. 80, and fixed capital sold to section I, i.e. 53.33, so that $Y_I = 133.33$.

We can check that the total price of fixed capital given by equation (41) is indeed:

$$P_I = L_I(1 + \alpha + \gamma^* + \alpha\gamma^*) = 50(1 + 0.667 + 0.6 + 0.4) = 50 \times 2.667 = 133.33$$

$$\text{Thus we have } Y = Y_I + Y_{II} = 133.33 + 200 = 333.33$$

We can also check that this last amount is equivalent to that given by equation (43) as being:

$$Y = L_{II}[(1 + \gamma) + k(2 + \alpha + \gamma + \gamma^* + \alpha\gamma^*)] =$$

$$Y = 100[(1 + 0.333) + 0.5(2 + 0.667 + 0.333 + 0.6 + 0.4)] = 100(1.333 + 0.5 \times 4)$$

$$Y = 100 (1.33 + 2) = 100 \times 3.33 = 333.33$$

This is indeed the case, which by the same token validates the formula.

## 4.2. The share of wages and profits in the product

The share of wages in section I over the price of fixed capital produced in this section is given by the initial expression of equation (45), and equal to:

$$\frac{W_I}{P_I} = (1 - a)(1 - c^*) = 0.6 \times 0.625 = 0.375$$

We can check that it is indeed equal to:

$$\frac{W_I}{P_I} = \frac{50}{133.33} = 0.375$$

The share of wages in section II over the price of all consumption goods produced in this section is given by equation (44) and equal to:

$$\frac{W_2}{P_{II}} = \frac{1 - c}{1 + k} = \frac{1 - 0.25}{1 + 0.5} = 0.5$$
We can check that it is indeed equal to:

$$ \frac{W_2}{P_{II}} = \frac{L_{II}}{L_{II}(1+k_c)} = \frac{100}{(100)(1+1)} = \frac{1}{2} = 0.5 = 50 \% $$

The share of wages in the product, i.e. over the total price of this product, is given by equation (46) and such as:

$$ \frac{W}{Y} = \frac{1+k}{1+\gamma+k(2+\alpha+\gamma+\gamma^*+\alpha\gamma^*)} $$

This formula may seem rather complicated, but we know that $k = 0.5$ and that the values of parameters $\alpha$, $\gamma$ and $\gamma^*$ are the following:

$$ \alpha = \frac{a}{1-a} = \frac{0.4}{0.6} = \frac{2}{3} = 0.667 ; \gamma = \frac{c}{1-c} = \frac{0.25}{0.75} = \frac{1}{3} = 0.333 ; \gamma^* = \frac{c^*}{1-c^*} = \frac{0.375}{0.625} = 0.6 $$

Therefore the expression above can easily be calculated as:

$$ \frac{W}{Y} = \frac{1+0.5}{1+0.333+0.5(2+0.667+0.333+0.6+0.4)} = \frac{1.5}{1.333+0.5(4)} = \frac{1.5}{3.333} = 0.45 $$

Moreover, we can check that this amount is exactly equivalent to the ratio of the overall wages $W = 150$ over the price of the product $Y = 333.33$

As for profits, in section II their amount is $\Pi_{II} = L_{II}k_c = 100 \times 1 = 100$, corresponding to the amount already found by difference between the price of consumption goods and wages in this section. We already know that their share in the product of Section II is equal to:

$$ \frac{\Pi_{II}}{P_{II}} = \frac{L_{II}k_c}{L_{II}(1+k_c)} = \frac{k_c}{1+k_c} = \frac{1}{2} = 50\% $$

Profits in section I were already calculated in the subsection 3.2, as being $\Pi_I = 83,33$ (see above p. 263) and also the price of section I product, as being $P_I = 133,33$ (see above p. 264).

Hence their share in the product of section I is the ratio $\frac{\Pi_I}{P_I} = \frac{83.33}{133.33} = 0.625$

We can check that it is their share in the product of section I calculated from equations (48) and (40):

$$ \frac{\Pi_I}{P_I} = \frac{L_I(\alpha + \gamma^* + \alpha\gamma^*)}{L_I(1 + \alpha + \gamma^* + \alpha\gamma^*)} = \frac{\alpha + \gamma^* + \alpha\gamma^*}{1 + \alpha + \gamma^* + \alpha\gamma^*} = \frac{1.667}{2.667} = 0.625 $$
Total profits for both sections are therefore 100 + 83.33 = 183.33, which corresponds exactly to the amount that can be calculated from the formula of equation (51):

$$ 
\begin{aligned}
\Pi &= L_{II} [k(\alpha + \gamma^* + \alpha\gamma^*) + k_c] = 100[0.5(0.667 + 0.6 + 0.667 \times 0.6) + 1] = \\
\Pi &= 100[0.5(1.667) + 1] = 183.33
\end{aligned}
$$

The share of profits in the total product is thus: $\frac{\Pi}{Y} = \frac{183.33}{333.33} = 0.55 = 55\%$ and we can check that it is equal to the ratio given by the developed formula of equation (52):

$$ 
\begin{aligned}
\frac{\Pi}{Y} &= \frac{\gamma + k(1 + \alpha + \gamma + \gamma^* + \alpha\gamma^*)}{1 + \gamma + k(2 + \alpha + \gamma + \gamma^* + \alpha\gamma^*)} = \\
\frac{\Pi}{Y} &= \frac{0.33 + 0.5(1 + 0.667 + 0.333 + 0.6 + 0.4)}{1 + 0.33 + 0.5(2 + 0.667 + 0.333 + 0.6 + 0.4)} = \frac{0.33 + 0.5 \times 3}{1.33 + 0.5 \times 4} = \frac{1.83}{3.33} = 0.55
\end{aligned}
$$

### 4.3. The profits/wages ratio

It is already known that $\pi_{II}$, the ratio of profits to wages in section II, is equal to the rate of surplus-value $k_c$. Thus we have $\pi_{II} = \frac{\Pi_{II}}{W_{II}} = \frac{100}{100} = 1 = 100\%$

We can also calculate the ratio of profits over wages in section I, which we call $\pi_I$.

$$ \text{This share is equal to } \pi_I = \frac{\Pi_I}{W_I} = \frac{83.33}{50} = 1.667 = 167\% $$

We can check that it is exactly the same as the share given from the more general formula of equation (53):

$$ 
\begin{aligned}
\pi_I &= \frac{\Pi_I}{W_I} = \frac{L_I(\alpha + \gamma^* + \alpha\gamma^*)}{L_I} = \alpha + \gamma^* + \alpha\gamma^* \\
\pi_I &= 0.667 + 0.6 + (0.667 \times 0.6) = 1.667 = 167\%
\end{aligned}
$$

Finally, as for the ratio of total profits over total wages, which is called $\pi$, it is:

$$ \pi = \frac{\Pi}{W} = \frac{183.33}{150} = 1.222 = 122\% $$

We can again check that it corresponds to the general formula of equation (55):

$$ \pi = \frac{k(1 + \alpha + \gamma + \gamma^* + \alpha\gamma^*) + \gamma}{1 + k} = $$
$$ \pi = \frac{0.5(1+0.667+0.333+0.6+0.4)+0.333}{1.5} = \frac{0.5 \times 3 + 0.333}{1.5} = 1.222 = 122 \% $$

We have thus verified that this model, as it was constructed, actually allows us to understand the realization of the totality of the product of the two sections, when we introduce the consumption of capitalists.

However, this product is realized according to specific modalities that differ in each of the two sections. In section II, profits corresponding *inter alia* to this consumption are obtained from the sale of consumption goods to workers of each of the two sections, as well as to capitalists of both sections. In section I, profits are obtained first from the sale of fixed capital to capitalists of section II and then from the sale of the remaining fixed capital within section I itself. These profits are therefore logically formed from the redistribution to section I of the surplus-value initially realized under the form of monetary profits in section II, which are then disseminated within section I.

Having completed the exploration of this model, we can now develop a few teachings that can be derived from it, regarding the significance of distribution and prices. This will be done in the next chapter.
# Chapter 14. Distribution and prices: a close connection

To be sure, the model with capitalist consumption developed in the last chapters has the capacity to shed some light on the main characteristics of distribution and prices in the capitalist mode of production. But it relies on a special case, i.e. a particular assumption regarding the fraction of the fixed capital available and bought at each stage by capitalists in section I, after previous sales of fixed capital have been made: this assumption is the fixity of this parameter, which was called $u$ in sub-section 3.3. of chapter 11, even if it was different from $b$. It seems therefore useful to get rid of this assumption, in order to get a more realistic view of distribution. We will first expose this generalization of the model with capitalist consumption, which will then enable us first to better understand how microeconomic prices are set, and second the process of transformation of values into monetary prices.

## 1. A generalization of the model with capitalist consumption

In order to examine a more general situation, as a first step it is necessary to reintroduce this parameter $u$ in a model with capitalist consumption. This makes it possible to establish a new table, named 14.1, with $u \neq b$ and therefore $u \neq 1-a$. This explains why the first row of the table, which shows the sale of fixed capital to capitalists of section II – where the share of fixed capital by definition stays as $u_1=b=1-a$, is not homogeneous to the others.

<table><caption>Table 14.1 - Prices, money values and profits in section I, with capitalist consumption</caption><thead><tr><th>Realized values = labor price = wages (w = 1)</th><th>Price of the product</th><th>Levied values = realized profits</th><th>Profits available for consumption</th></tr></thead><tbody><tr><td>(1 - a) L<sub>I</sub> = b L<sub>I</sub></td><td>(1 - a)L<sub>I</sub> / (1 - a) &middot; (1 - c*) / (1 - c*) = L<sub>I</sub> / (1 - c*)</td><td>L<sub>I</sub> (a + c*) / (1 - c*)</td><td>(1 - a)L<sub>I</sub> / (1 - c*)</td></tr><tr><td>auL<sub>I</sub></td><td>a L<sub>I</sub> / (1 - c*)</td><td>aL<sub>I</sub> (1 / (1 - c*) - u)</td><td>auL<sub>I</sub> / (1 - c*)</td></tr><tr><td>au(1 - u) L<sub>I</sub></td><td>a(1 - u)L<sub>I</sub> / (1 - c*)</td><td>a(1 - u)L<sub>I</sub> (1 / (1 - c*) - u)</td><td>au(1 - u)L<sub>I</sub> / (1 - c*)</td></tr><tr><td>au(1 - u)<sup>2</sup> L<sub>I</sub></td><td>a(1 - u)<sup>2</sup> L<sub>I</sub> / (1 - c*)</td><td>a(1 - u)<sup>2</sup> L<sub>I</sub> (1 / (1 - c*) - u)</td><td>au(1 - u)<sup>2</sup> L<sub>I</sub> / (1 - c*)</td></tr><tr><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>au(1 - u)<sup>n</sup> L<sub>I</sub></td><td>a(1 - u)<sup>n</sup> L<sub>I</sub> / (1 - c*)</td><td>a(1 - u)<sup>n</sup> L<sub>I</sub> (1 / (1 - c*) - u)</td><td>au(1 - u)<sup>n</sup> L<sub>I</sub> / (1 - c*)</td></tr></tbody></table>
Nevertheless, and to begin with, we introduce $u$ as a constant, as the table shows, just as a first step allowing for an easier understanding of this matter. We must also note that the parameter which corresponds at each stage to the ratio of profits over product price, which is usually called the gross profit margin, and is obtained by dividing for each row of the table the profits in the third column by the product price in the second column, is no longer $1-u$ (like in table 3 of chapter 11), but becomes now $1-u(1-c*)$. The detail of the calculation of each term is not repeated, since this above table 8 is in fact nothing else than a combination of table 6 of chapter 11 and table 7 of chapter 12.

If we want to check the validity of this table, we can make a quick test by calculating the sum of prices for all of the fixed capital produced and sold by section I. It is obtained by summing the quantities in column 2 of the table:

$$P_I = \frac{L_I}{1-c^*} \left[ 1 + a \left\{ 1 + (1-u) + (1-u)^2 + \dots + (1-u)^n \right\} \right]$$

$$P_I = \frac{L_I}{1-c^*} \left[ 1 + a \left( \frac{1}{1-(1-u)} \right) \right]$$

Knowing that $1-u=1-b=a$, $P_I$ becomes:

$$P_I = \frac{L_I}{1-c^*} \left[ 1 + \frac{a}{u} \right] = \frac{L_I}{1-c^*} \left[ 1 + \frac{a}{1-a} \right] = \frac{L_I}{1-c^*} \left( \frac{1}{1-a} \right) = \frac{L_I}{(1-c^*)(1-a)} \quad [QED] \quad (1)$$

Indeed we have $P_I = I = Y_I$, i.e. exactly the price of fixed capital given by equation (39) of chapter 12.

Before going further, we must recall that the exogenous parameters that we defined as such in the previous chapters must be considered as invariant at least at a macroeconomic level within the framework of a given reproduction scheme. These parameters are $k$, $a$ (and therefore $b=1-a=u_1$), $P_{II}$ from which we derive $c$ (the share of capitalist consumption in total consumption), and $c^*$ (the parameter linked to capitalist consumption in section I).

In a system generalizing the scheme provided in table 8 on the previous page, we must now consider that, outside the sale of fixed capital from section I to section II, whose amount is fixed and exogenous, parameter $u$ is different for each stage of fixed capital sales within section I: in other words there are as many $u_i$ as there are such stages, with the first one being eliminated from the first row (corresponding to the sale of fixed capital to section II) as being equal to $b=1-a$, when it is divided by $(1-a)$ in the expression of the price. This first $u$ (or $u_1$) is thus fixed by the technological constraint that section II gets in any case from section I a share $b=1-a$ of the total value of fixed capital.
The multiplicity of the other $u_i$ parameters makes the various terms in the table become more complex. All the more so that there is also no reason why capitalist consumption should be the same at each stage: in the real world there is on the contrary every reason to think that the fraction of the value of consumption goods obtained at each stage by capitalists will be different from one stage to another.

Obviously this multiplicity introduces more complexity in the process of price fixation and product realization, but it is nevertheless the price to pay for a better understanding of the real world. Therefore, instead of having just one fraction $c^*$ corresponding to $c_I$, i.e. the share of global consumption obtained by all of the capitalists of section I, we will have as many $c_1^*$, $c_2^*$, ..., $c_n^*$ fractions as there are stages. However the overall consumption of capitalists in section I, as an exogenous parameter, must stay at a level $c_I$ which remains a constant (and thus $c^* = c_1^*$) as it appears in the first row of the above table). With these new assumptions regarding $u$ and $c^*$ we can thus explore what will happen of prices, taking into account this additional complexity.

This in fact comes down to leaving the universe of macroeconomic prices, where we have been evolving so far, i.e. prices of two broad categories of commodities (consumption goods and fixed capital) linked by global constraints, and going to the area of microeconomic prices, i.e. prices such that one price corresponds theoretically to one single commodity.

## 2. From macroeconomic to microeconomic prices

### 2.1. Prices of fixed capital

To continue with prices of fixed capital, and starting therefore with them, let us explore the consequences of having different values at each stage of exchanges within section I for parameters $u$ and $c^*$. While doing that, we must consider first that the overall price of consumption goods produced by section II obviously does not change.

Similarly, the price of fixed capital sold to this section II (in the first row of the table) will not change because of changes in the value of parameter $u$, since this parameter $u_1$ has been defined as a share of values, and not as a share of prices, and as we have just indicated is constrained at this stage by representing in any case the exogenous share $b=1-a$ of the value of fixed capital going to section II: $u_1=b=1-a$.

The prices that will change because of differing parameters $u$ are the prices for the sales of fixed capital within section I from the second row and onward. Moreover, capitalists selling fixed capital to section II (first row) will now obtain a fraction of the value of total consumption depending on $c_2^*$, capitalists selling fixed capital to them a fraction depending on $c_3^*$, and so on. Therefore all of the prices of fixed capital goods have to adjust because of
changes in the fraction of the overall value of consumption goods which capitalists selling fixed capital to other ones are able to get, by adjusting their prices.

Thus, although the price of fixed capital sold to section II, appearing in row 1 stays as $\frac{L_I}{1-c^*}$, the price for fixed capital in the second row, instead of $a \frac{L_I}{1-c_2^*}$ becomes $(1-u_2) \frac{L_I}{1-c_2^*}$.

For row 3 of the table, the price thus becomes $(1-u_2)(1-u_3)L_I \frac{1}{1-c_3^*}$, and for row 4 $(1-u_2)(1-u_3)(1-u_4)L_I \frac{1}{1-c_4^*}$, and so on and so forth. The general formula for the price of fixed capital of the m^th stage appearing in row m can then be written:

$$ \text{Price of fixed capital (row m)} = \left[ \prod_{i=2}^{i=m} (1-u_i) \right] L_I \frac{1}{1-c_m^*} \quad (2) $$

As for the realized money value of fixed capital, it stays unchanged as $(1-a)L_I = bL_I$, as far as fixed capital sold to section II is concerned, in the first row of the table. But it becomes $au_2L_I$ for the money value of fixed capital in the second row sold to the previous stage, and $au_3(1-u_2)L_I$, in the third row. For the fourth row it becomes $au_4(1-u_2)(1-u_3)L_I$, and so on and so forth. The general formula for the realized value of fixed capital of the m^th stage appearing in row m can then be written:

$$ \text{Realized value of fixed capital (row m)} = au_m \prod_{i=2}^{i=m} (1 - u_i)L_I \quad (3) $$

In this more general system, instead of having at each stage $b=1-a$ as a fixed parameter, we have $b=u$ as a variable parameter, meaning in other words that at each stage the share of the remaining value of fixed capital which is obtained is not predetermined. It follows that the various fixed ratios which we had defined in chapter 11 become variable ones:

$$ \text{The ratio of sale price over money value (wages) becomes: } \frac{1}{u_i(1-c_i^*)} \quad (4) $$

$$ \text{Conversely the ratio of money value (or cost price) over sale price becomes: } u_i(1-c_i^*) \quad (5) $$

$$ \text{Thus the gross profit margin (or ratio of profits over sale price) becomes: } 1-u_i(1-c_i^*) \quad (6) $$
The ratio of profits over money value (or cost price) becomes: $\pi_i = \frac{1}{u_i(1-c_i^*)} - 1$ (7)

In this more general case one might wonder whether all of these successive exchanges of fixed capital sold with changing margins $1-u_i(1-c_i^*)$ at each stage $i$ will make it possible to clear the whole product of section I.

Indeed the sum of prices of fixed capital sold at each level is:

$$I = \frac{L_I}{1-c_1^*} + (1-u_2)\frac{L_I}{1-c_2^*} + (1-u_2)(1-u_3)\frac{L_I}{1-c_3^*} + \dots + \prod_{i=2}^{i=n}(1-u_i)\frac{L_I}{1-c_n^*} \quad (8)$$

Whereas we know from equation (39) that the sale of the whole fixed capital corresponds to a situation where for macroeconomic reasons linked to simple reproduction ($a$ and $c^*$ being macroeconomic parameters) we have:

$$I = Y_I = I_I + I_{II} = \frac{L_I}{(1-a)(1-c^*)} \quad (9)$$

To be sure this creates a macroeconomic constraint on the values of all $u_i$ and $c_i^*$, but there is no particular reason to think that there would not be at least one system of values for these variables which would not meet the objective of selling the whole fixed capital. The reason to believe that such a situation may exist is that the system is quite flexible. First, because each stage depends on the previous one, which implies that at each stage there is a possibility to adjust the corresponding variables to the situation already achieved at the previous stage. And second because there is an important element of flexibility embodied in the system: as indicated previously all of $c_i^*$, which define the consumption of capitalists at each stage, correspond to residual levels of consumption. In a situation of simple reproduction corresponding to a self-replacing state the variation of consumption levels would thus play the role of the adjustment parameter which would allow to meet the desired level of investment, and thus to absorb all the production of fixed capital.

Finally there is a last element of flexibility coming from the fact that at each stage we are still in the real world at a macroeconomic level, because in the real world each stage corresponds to a number of different commodities composing fixed capital, each commodity with its own price. This multiplies the number of different parameters $u_i$ and $c_i^*$, but also the possibilities of adjustment.

This brings us to the question of microeconomic prices for consumption goods produced in section II.
## 2.2. Prices of consumption goods

In the real world there are a lot of consumption goods and therefore their total price given by equation (12) in chapter 11 cannot but be anything else than an average.

We recall that this total price is $P_{II} = L_{II}(1+k_c) = L_{II}\left(1+\frac{k+c}{1-c}\right) = L_{II}\left(\frac{1+k}{1-c}\right)$

To be sure $k$ is a technological constant, but it plays this role only at macroeconomic level, and it is the same for $c$. At microeconomic level there is no particular reason why $k$ and $c$ should be the same for whatever consumption good, as long as their level allows for the desired level of investment in fixed capital and for a residual amount of consumption.

It follows that the various fixed ratios which can be derived from this general formula become variable ones:

The ratio of sale price over money value (wages) becomes: $\frac{1+k_i}{1-c_i}$ (10)

Conversely the ratio of money value (or cost price) over sale price becomes: $\frac{1-c_i}{1+k_i}$ (11)

Thus the gross profit margin (i.e. profits over sale price) becomes: $\frac{k_i+c_i}{1+k_i}$ (12)

The ratio of profits over money value (wages or cost price) becomes: $\frac{k_i+c_i}{1-c_i}$ (13)

Therefore for each price the margin over the money value (wages) which is given by this last equation can be variable from one consumption good to another, knowing that the weighted average for all goods (the weights being the respective values) must stay at $\frac{k+c}{1-c}$, i.e. the predetermined rate of surplus-value, which is a fundamental parameter of the whole economic system, as a key for understanding distribution and the modalities of its reproduction.

The constraint created by this average must not be underestimated, because it means that for the reproduction of the system to take place, then the sum of prices of consumption goods (weighted by their respective values), must be such that $P_{II} = L_{II}\left(\frac{1+k}{1-c}\right)$. For given levels of $k$ and $c$, this creates an interdependence between all of the prices of consumption goods.

## 2.3. Prices of intermediate goods

So far, because it was in the logics of a model concerned with problematics of reproduction, we dealt only with final goods, be they consumption goods or fixed capital goods. The price
of intermediate goods, comprising both wages and a profit margin included in their sale price, was supposed to be included in the price of final goods: the amount of wages (or money value) in the price of final goods therefore included the wages spent in the production of these intermediate goods as well as the profit margin contained in their sale price to final producers.

But intermediate goods must necessarily be introduced, if only because they can play an important role at macroeconomic level, by influencing both distribution and the price of final goods. Recalling the role played by the price of oil at world level is enough to justify that.

However it seems quite difficult, for the determination of their price, to devise a general formula similar to that of consumption goods or fixed capital goods, if only because a good number of intermediate goods are not specific to only one of both sections of the productive system, but on the contrary can be used simultaneously by both sections, to produce consumption goods as well as capital goods. Their influence on the final price of goods may also vary greatly from one intermediate good to the other, depending *inter alia* on their substitutability.

At any rate this influence depends on a case by case microeconomic analysis, which might imply sometimes to begin with a mesoeconomic one (like for oil products), which is not the object of the present work. It is the reason why we prefer to make the simple assumption that for each particular intermediate good its price is simply the sum of its money value (wages) and of a profit margin that we will call $\pi_i$, and which is given as an exogenous data for each good. As it will be shown now this will be sufficient to go from values to prices.

# 3. From values to prices

## 3.1. A few calculations of values and prices

### 1) The value system

Let us first consider that there are $n$ commodities, and that their values are the elements of a vector of values $V = [v_1 \dots v_n]$. Since it is a row vector these elements are named $v_j$ (with $j = 1, \dots, n$). We continue to define the value $v_j$ of one unit of commodity $j$ as the average social labor time it takes to produce commodity $j$. In order to transform values into prices without unnecessarily complicating the demonstration, we will assume that there are no intermediate commodities of the second type (as defined previously), and that we have $n$ commodities, i.e. $k$ intermediate commodities of the first type and $n-k$ final commodities.

Since commodity $j$ has been produced through the transformation of various other intermediate commodities, $v_j$ can be defined first as the sum of the direct labor time $l_j$ needed to produce commodity $j$, plus the indirect time corresponding to the value of the commodities that have been transformed to produce commodity $j$, which is the value of its means of production, i.e. the intermediate commodities $i$ used in its production, with all of them supposed to be of the first type, and thus with $i = 1, \dots, k$.
$$ \text{Thus } v_j = a_{1j}v_1 + a_{2j}v_2 + \dots + a_{jj}v_j + \dots + a_{kj}v_k + l_j \quad \text{or} \quad v_j = \sum_{i}^{k} a_{ij}v_i + l_j \quad (14) $$

In chapter 6 we had already determined a set of values for commodities, with a second method based on a system of two matrix equations (see section 3 of chapter 6).

Let us name $V_1$ the row vector of values of intermediate goods and $V_2$ the row vector of values of final goods, $A_{11}$ and $A_{12}$ the matrices of intermediate coefficients, $L_1$ and $L_2$ the vectors of quantities of direct labor for the production of intermediate goods and final goods respectively. The matrix equations that we can write for values are:

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

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

Which gives us, as solutions to these equations:

$$ V_1 = L_1 (I - A_{11})^{-1} = L_1 (I + A + A^2 + \dots + A^n) \quad (17) $$

$$ V_2 = L_1 (I - A_{11})^{-1} A_{12} + L_2 \quad (18) $$

## 2) The price system

To go from this system of average social values to a price system, we first have to transform it into a monetary system of money values, which only implies to set as the unit linking both systems the level of the average monetary wage per physical time unit of labor time, i.e. $\bar{w} = 1$ (knowing that $\bar{w}$ itself is not a scalar but has the dimension of a scalar divided by a time). Multiplying values by $\bar{w} = 1$ gives us the system of money values which is made of pure scalars and can be considered as the mirror of the former, and thus can be written exactly with the same notations. Thus the equations do not change, only the unit changes: it is a unit of physical time for the first system of average social values, and a unit of money for the system of money values, with $\bar{w}$ as the transformer.

The second thing that we need to do in order to determine the price system is to introduce profit margins, which cannot be determined at macroeconomic level, although they are submitted to macroeconomic constraints, but as we just indicated have to be provided by mesoeconomic and microeconomic analysis.

If we were doing that within the framework of standard economic theory, we would have first to use our profit margins to calculate profit rates, by dividing each amount of profit appearing in the production of a commodity by the amount of fixed capital used in its production. But we have not introduced so far the amount of the stock of accumulated fixed capital in our theoretical framework, because in fact this was not needed. As a kind of fallback solution we could consequently use the same formulation as Marx and Sraffa and decide to apply the
profit margin to the stock of circulating capital. Indeed Marx considered that circulating capital was a part of what he called constant capital, and that it appeared in the denominator of the expression giving the profit rate. As for Sraffa, in most of “Production of commodities...” he calculates the profit rate only over the cost of circulating capital made of basic commodities, and thus the expression giving the price of a particular intermediate commodity is:

$$p_j = (a_{1j}p_1 + a_{2j}p_2 + \dots + a_{jj}p_j + \dots + a_{kj}p_k)(1+r_j) + l_j w_j \quad (19)$$

However using this definition of prices would not be in line with the necessities of the reproduction of the system, as they were exposed in the previous chapters, which showed that the amount of profits and thus the profit margin incorporated in the price of each particular good must be such as to allow for each particular producer to buy all the fixed capital needed to carry out the production process of each particular good, on the one hand, and to buy some share of consumption goods available on the market, on the other hand.

These considerations imply that the expression giving the price of a particular intermediate commodity should rather be:

$$p_j = (a_{1j}p_1 + a_{2j}p_2 + \dots + a_{jj}p_j + \dots + a_{kj}p_k) + l_j w_j (1 + \pi_j) \quad (20)$$

Determining the price system implies to know all of the various profit margins $\pi_j$ (we must insist on the fact that they are not profit rates). Each one of these margins corresponds to the production of one particular good. We know that these $\pi_j$ are exogenous data, and as such they can be integrated in a system of matrices defining the price system. We just need first to define two simple matrices, one for intermediate goods and the other for final goods.

The first matrix concerns the $k$ intermediate goods: it is a diagonal matrix (and thus a square one) of dimensions $(k,k)$, named $\Pi_1$, in which the elements $\pi_i$ of the diagonal correspond to the profit margins over wages in all the different branches which produce these intermediate goods. This matrix is therefore of the general form:

$$\Pi_1 = \begin{bmatrix} \pi_1 & 0 & \dots & 0 \\ 0 & \pi_2 & \dots & 0 \\ \dots & \dots & \dots & \dots \\ 0 & 0 & \dots & \pi_k \end{bmatrix} \quad (21)$$

The second matrix concerns the $n-k$ final goods. It is also a diagonal matrix of dimensions $(n-k, n-k)$, named $\Pi_2$, in which the elements of the diagonal are $\pi_i = \frac{1}{u_i(1-c_i^*)} - 1$ for
fixed capital goods - see equation (7), and $\frac{k_i+c_i}{1-c_i}$ for consumption goods - see equation (13),
i.e. the respective profit margins over cost prices in both sections. It is of the general form:

$$ \Pi_2 = \begin{bmatrix} \pi_{k+1} & 0 & \dots & 0 \\ 0 & \pi_{k+2} & \dots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \dots & \pi_n \end{bmatrix} \qquad (22) $$

If $l_j w_j$ is expressed in units of average social labor multiplied by $\bar{w}$ (the average nominal wage), then it is possible to replace $L_1$ by $W_1$ and $L_2$ by $W_2$, just to emphasize the fact that we are in the domain of monetary variables.

The price system can therefore be determined without particular difficulties. Let us name $P_1$ the row vector of prices of intermediate goods, and $P_2$ the row vector of prices of final goods. $P_2$ combines the prices of fixed capital goods $P_I$ and of consumption goods $P_{II}$:

$$ P_2 = (P_I, P_{II}) $$

Then on the basis of equation (17) we can write:

$$ P_1 = A_{11} P_1 + W_1 (I + \Pi_1) \qquad (23) $$

$$ P_1 = W_1 (I - A_{11})^{-1} (I + \Pi_1) \qquad (24) $$

$$ P_1 = W_1 (I - A_{11})^{-1} + W_1 (I - A_{11})^{-1} \Pi_1 \qquad (25) $$

Equation (23) can be written:

$$ P_1 = W_1 (I + A_{11} + A_{11}^2 + \dots + A_{11}^n) + W_1 (I + A_{11} + A_{11}^2 + \dots + A_{11}^n) \Pi_1 \qquad (26) $$

Before writing the equation giving the price of final goods we can observe that the first member on the right side of equation (25) provides us with the total amount of wages paid for the production of intermediate goods. For each good, we can get the amounts of direct wages paid for labor directly employed in the production of the various goods (i.e. $W_1 I$) and for the production of intermediate goods that are used at the next levels: $W_1 A_{11}$, $W_1 A_{11}^2$, and so on. For a given commodity, it is this total amount which would correspond to the amount of wages appearing in the equations of the reproduction scheme of the previous chapters.

We have also defined profit margins as appearing at each level of the production system, which implies that a profit margin at one level only applies to wages spent at this level, which are in any case the only wages known at this level. Therefore profits at one level cannot be
calculated on wages at another level. The implications for writing the equation giving the price of final goods are that wages and profits realized at the last level of their production as final goods are additive to profits and wages which are part of the price of intermediate goods transformed into these final goods. Then on the basis of equation (18) we can write:

$$P_2 = P_1(A_{12}) + W_2(I + \Pi_2) \quad (27)$$

Combining equation (24) with equation (27), we obtain:

$$P_2 = [W_1(I - A_{11})^{-1} A_{12}] (I + \Pi_1) + W_2(I + \Pi_2) = \quad (28)$$

$$P_2 = [W_1(I - A_{11})^{-1} A_{12} + W_2] + [W_1(I - A_{11})^{-1} \Pi_1 A_{12}] + W_2 \Pi_2 \quad (29)$$

The last equation (29) seems somewhat complicated, but it means only that the price of final goods can always be reduced to a sum of wages and profits:

1) First a sum of wages, i.e. $W = W_1(I - A_{11})^{-1} A_{12} + W_2$, comprising:

a. Wages paid indirectly in the production of intermediate goods transformed into final goods: $W_1(I - A_{11})^{-1} A_{12}$,

b. Wages paid at the last stage of production of these final goods: $W_2$;

2) Secondly a sum of profits:

a. Profits realized in the production of intermediate goods: $W_1(I - A_{11})^{-1} \Pi_1 A_{12}$

b. Profits realized at the last stage of production of final goods: $W_2 \Pi_2$

Because of the assumptions that we made, these profits are additive, and thus the overall profit margin over wages in the production of a particular final good can be considered as the weighted average of all profit margins over wages at all stages of production of intermediate goods transformed into this final goods as well as at the last stage of transformation into this final good. The weights are obviously the amounts of wages at each stage. This weighted average would correspond to the profit margins as defined in the previous chapters.

## 3.2. Some preliminary findings

What can be immediately seen from the final equations giving prices is the complete interdependence between prices and distribution.

1) First because prices are monetary prices and derive from values that are money values equivalent to wages, which themselves are a fundamental distribution variable, even though we have shown that wages depend on a number of parameters which are not of a purely
economic nature. It is indeed impossible to deny that the average level of nominal wage, a fundamental parameter of the economic system, as well as the hierarchy of these nominal wages among various categories of labor and workers, depend on a number of elements of an ideological, political, institutional and sociological nature, as well as others of an economic nature, like the level of labor productivity or the level of unemployment.

2) Second, because prices depend also on the level of profit margins which are to some extent arbitrary, and have important degrees of freedom. However, as soon as prices of consumption goods are set by capitalists producing these goods, the most fundamental variables of distribution are simultaneously determined: the share of the value of the whole product going to workers and to capitalists is fixed as the rate of surplus-value, as well as the share of the value of consumption goods going to capitalists (who in any case get by definition the whole value of fixed capital). Similarly, as soon as capitalists selling fixed capital goods set their price, first the share of the value of consumption goods going to both categories of capitalists (producing either consumption goods or fixed capital goods) is determined - this value of capitalist consumption itself having been predetermined at the previous step. Simultaneously setting the price of fixed capital goods and the profit margins to which these prices correspond also determines the sharing of consumption goods among capitalists producing fixed capital.

The only way to change this distribution of values among the various stakeholders is to change prices, which, for a given level of wages, implies to change the various profit margins, on consumption goods as well as on fixed capital goods.

Thus we are led to realize that distribution and prices are consubstantial: to change one changes the others and reciprocally. This makes us understand that the only economic variables which are totally invariant to changes in distribution are values, because values belong to another conceptual realm than prices, having been determined at the level of production. On the contrary prices, which cannot be determined as long as production has not taken place, are variables defined at the level of distribution and depending on distribution variables, i.e. the various levels of wages and profit margins, just as much as these distribution variables depend on prices.

It means that for a given system of production, to which corresponds one and only one set of values, there can be an infinity of possible distribution schemes, which distribute values among various stakeholders, each of these specific schemes being linked to a corresponding price system. Reciprocally each price system is linked to a specific distribution scheme. Therefore, without denying that production parameters obviously influence prices, they do that via values, and prices should be considered much more as distribution prices than as production prices. At a theoretical level, there is no such thing as production prices: there only are distribution prices, which is indeed a fundamental finding.

3) Third, it must be emphasized that it is only for intermediate goods that we have an equation corresponding to Sraffa's system of production prices for basic commodities, and
which uses the Leontief inverse matrix $(I - A_{11})^{-1}$. This shows that this method for calculating prices is only relevant for intermediate goods, to the extent that the level of money wages and the levels of margins over wages are given for each particular commodity. Indeed for Sraffa all final goods like consumption goods are luxury goods. As for fixed capital goods we have shown earlier that for Sraffa they are wrongly considered as not being final goods, but as a particular species of circulating capital. This specificity comes from their transferring their value (in fact their price) to products during more than one production cycle, but as we also showed Sraffa fails to introduce this special feature in his system in a non-contradictory way. We shall come back in more details on this distinction between various categories of commodities.

This use of the Leontief matrix is made possible only because in the equations giving values as well as prices for these intermediate goods we have the same goods and prices on both sides of the equations [see equations (15) and (23)], whereas for final goods their price appears only on the left side of the equations, and depends on other prices on the right side, i.e. precisely those of intermediate goods.

This should call for caution when using Leontief inverse matrix in systems with heterogeneous categories of commodities, as if all commodities could be considered as intermediate goods, which is not at all the case.

4) As a corollary to the previous finding, although it is obvious that there is production of fixed capital in our reproduction schemes, we have been able to devise both a value and a price system for commodities in a scheme of simple reproduction without this fixed capital transferring any value to the product. This did not prevent fixed capital to be sold, first to capitalists of section II producing consumption goods, and secondly to capitalists of section I producing fixed capital.

5) Finally, it is notable that the rate of profit does not appear in the equations defining prices, even for intermediate goods. This might seem weird, but derives from the fact that fixed capital is not an intermediate good and does not transfer its value to the product. This does not imply that we would be unable to define profit rates, and in fact we shall do that later. Indeed when we know profit margins we also know the absolute amounts of profits, and we could calculate the profit rate by dividing this amount by the price of the respective stocks of capital.

This shows nevertheless that the emphasis put by all economic literature (including Marx himself) on the rate of profit is grossly mistaken. We shall also come back to this point, but the fact that we do not need to know the profit rates to calculate prices draws our attention to the fact that we do not need either to know the price of the stock of fixed capital in order to determine the price system. It is true that the system determines the price of the various assets constituting fixed capital and which are produced during the period. It would also enable us to determine the price of the total fixed capital stock in operation, if we knew the lifetime of these various assets, as a technical data, and the price at which they were initially bought. But
# Chapter 15. A coherent theory of rent

This chapter elaborates on what has been demonstrated so far to expose a coherent theory of rent. The first section starts by recalling that there are two kinds of rent: differential rent and absolute rent. To illustrate how rent is fixed, it exposes four different systems of prices for the same commodity (wheat), produced on lands of different “fertilities”. The first system is characterized by zero global rent, the second one by zero total rent and equalization of prices by the State, the third one by a zero differential rent on marginal land, and the fourth one by a zero differential rent and a positive absolute rent on marginal land. The second section can thus analyze type II differential rent for Sraffa and Marx, i.e. the case of the production of wheat with different techniques on homogeneous land. All this allows in a third section to understand better the effect of rent on the price system and distribution.

## 1. The nature of rent, differential rent and absolute rent

Let us begin by recalling that rent is supposed to be an income corresponding to the "contribution to production" of the non-produced means of production, such as land, and that this definition includes other unproduced natural resources such as mines, or more precisely mineral deposits. In line with the model developed so far, the essential difference with fixed capital as a means of production is the unproduced character of these resources, which implies that it is impossible to calculate for them a cost of production as it is the case for machines. Another difference is that land is supposed to be immutable, and therefore has an infinite life span. But this difference does not apply to mines, which all end up being exhausted.

In Sraffa's book "Production of commodities...", there is an analogy between these unproduced means of production, such as land, on the one hand, and fixed capital, on the other hand, which consists in their presence on the left side of the equations defining the prices of production. But, unlike fixed capital, their absence on the right side of the system of equations does not, however, permit them to be included in the net standard product or in the composition of the standard commodity. This also explains why their price is calculated differently: fixed capital may appear among final goods on the right side of the equations defining prices, whereas land does not appear there, and its price is determined thereby from the rent it provides. Moreover, it is clear that there is no consumption of land in the production process, and therefore it is out of the question to treat it as an intermediate good, through any depreciation, which is not conceivable.

Finally, as we already saw, the price system is calculated by retaining, for each product for which land is used, the only equation in which this rent is zero, and where therefore land does not appear even on the left side of the equations. As has also been shown, for this reason, rent remains the stumbling-block of Sraffa's price system, with fixed capital - for other reasons, since the "fertility" of land varies with distribution and the system of prices, which makes it impossible to determine which is the land without rent without knowing first the price system! Once again, a theory is invalidated by the circularity of its reasoning.
These different characteristics of land, as well as our model, justify Marx when he points out that, as for fixed capital, the remuneration of land exists only because of its appropriation and more precisely because of the private property of land. This explains why in the real world rent is not zero even on the marginal land, where the cost of production is the highest, but where, nevertheless, there is a positive rent. This is what Marx calls absolute rent. If it did not exist, the landlord owning this marginal land would have no interest in renting it for cultivation. One could argue that this is due to the scarcity of land, and that if there is free land then the marginal rent must be zero. But this objection is based on a confusion over the meaning of the word "free", which means not only "not cultivated" but "not the object of appropriation". It does not take into account that precisely in the capitalist mode of production there is no free land, in the sense of land without owners. One cannot, therefore, cultivate even a "free" land, but this time in the sense of a land "not cultivated so far", and thus such as scarcity cannot manifest itself, unless a rent is paid to its owner.

Another proof of the fact that rents remunerate the ownership of unproduced means of production, and not their "productivity", comes from the fact that land, like fixed capital, does not create value: it transmits no value to the product. If land transmitted value, it is difficult to see why the marginal land would not transmit it, especially since with the extension of cultivated areas, a marginal land at a given moment ceases necessarily to be a marginal one later. In fact, like fixed capital, land is a catalyst, and in this case it is the support, or rather the substratum of the production process for agricultural products. This support can even be dispensed with in the case of so-called landless production, like hydroponic production, realized without any land. It follows that rent, as a remuneration for the ownership of land, is just like profits taken also from surplus-value, and consequently follows identical mechanisms.

In the model which has been described hitherto, and as we shall now show, the introduction of rent no longer runs up against the difficulties encountered in Sraffa's system: there is no need to know first the land without rent, in order to determine only after that - and simultaneously, distribution and prices. To introduce rent into our system, we just need to start by assuming that there is only one commodity, e.g. wheat, produced with lands of different "fertility", and with different quantities and proportions of circulating capital, considering at first that there is no rent. In order to introduce rent, we will proceed in stages, which will be done by assuming that there are always *n* kinds of lands of different "fertility", all producing wheat with different production processes, and by examining successively four systems of equations in which rent exists but manifests itself in different ways.

## 1.1. The price of wheat with a zero total rent

We know that if we want to determine values and prices through systems of equations there can be only one single value and one single price for the same commodity, whatever the conditions of production. There can be only one equation giving the value of wheat, otherwise the system would be overdetermined. But this equation cannot in any case be the equation giving the conditions of production on the marginal land. This is true for the calculation of values, because it is the totality of the labor expended to produce wheat that must be taken
into account, and not only that expended on the marginal land, which, moreover, is not
known, since its determination depends on the price system. This is true also for the
calculation of prices, because otherwise one would fall back into a circular reasoning: the
system of equations allowing the calculation of prices assumes that the land without rent is
known, but in order to know what is this land without rent, one must know the price system!

Therefore, the equation giving the price of wheat must necessarily take into account all the
intermediate commodities and labor inputs used on all the lands simultaneously under
cultivation to produce the total quantity of wheat. On this basis, we can calculate the value of
wheat, like that of any other commodity, using equation (17) in the previous chapter:

$$
V_1 = L_1 (I - A_{11})^{-1} \quad \text{where the value of wheat is one of the values, e.g. } v_w \quad (1)
$$

Similarly, assuming that wheat is an intermediate commodity, its price can be calculated using
equation (23) in chapter 14 (see p. 278), where – for the sake of comparison with Marx and
Sraffa, and since it does not change the reasoning in any way, π as the profit margin is
replaced by r as the corresponding profit rate, this profit rate being moreover applied to the
price of circulating capital:

$$
P_1 = W_1 L_1 + A_{11} P_1 (I + R_1) \tag{2}
$$

In which the price of wheat is one of these prices, for example $p_w$ :

$$
p_w = \bar{w}l_w + a_w p_1 (1 + r_w) \quad (3)
$$

$p_1$ and $a_w$ are vectors, $\bar{w}$ is the average nominal wage, taken as the wage unit ($\bar{w} = 1$), $l_w$ is a given quantity of average social labor and $r_w$ is the average profit rate for the whole wheat producing branch.

This price of wheat will then and necessarily be an average price, like all other prices $p_i$. It
should be kept in mind that each equation of the price system does not reflect the conditions
of production of a given firm, but those of a whole branch, consequently composed of all
enterprises producing together the total quantity of each commodity, each with its own
production methods and techniques, which have no reason to be the same as those of all the
other firms producing the same commodity.

At the enterprise level, firms therefore have different conditions of production, which means
that they do not realize the average rate of profit, but a specific rate of profit, which may be
higher or lower than the average rate of profit. In the case of commodities for which land is
used as a means of production, the difference in production conditions and therefore in
production costs may originate firstly from the heterogeneity of land, which for the same area,
with the same amounts of circulating capital and labor, and with a given rate of profit,
produce different quantities, and therefore have a different yield and fertility. This is even
more true if the amounts of circulating capital and labor differ, as is generally the case.
For a given average price of wheat $p_w$ and average rate of profit $r_w$, the most "fertile" land with the lowest production cost will thus benefit from a rent corresponding to what Marx calls the differential rent of type I. But there is also a situation where production conditions may differ and the cost may be higher or lower per unit of quantity produced due to the use of different production techniques on a homogeneous land. This will make appear - for the firms with the lowest cost of production on the same land, what Marx considers to be also a rent, which he calls differential rent of type II.

This phenomenon can be translated into equations because, once prices are known, including the average unit price of wheat $p_w$ and the average rate of profit $r_w$ in the branch producing wheat, the equation giving the average price of wheat can be disaggregated within the price system, making land appear, and thus we can write as many equations as there are lands of different "fertility", so that we have, for $n$ lands of distinct "fertility":

$$ 
\begin{aligned}
\bar{w}_1 + a_1 p_1 (1 + r_w) + \rho_1 \Lambda_1 &= p_w \\
\bar{w}_2 + a_2 p_1 (1 + r_w) + \rho_2 \Lambda_2 &= p_w
\end{aligned}
$$

$$ \bar{w}_{l_g} + a_g p_1 (1 + r_w) + \rho_g \Lambda_g = p_w \quad \text{with } \rho_g = 0 \quad (\text{System 1}) $$

$$ 
\begin{aligned}
\bar{w}_{n-1} + a_{n-1} p_1 (1 + r_w) + \rho_{n-1} \Lambda_{n-1} &= p_w \\
\bar{w}_n + a_n p_1 (1 + r_w) + \rho_n \Lambda_n &= p_w
\end{aligned}
$$

In this system $a_i$ and $p_1$ which are known vectors, $\Lambda_i$ are the quantities of land of different fertility in units of area (hectare for example), and $\rho_i$ is the rent per unit of area. It should be emphasized that the system has been standardized to show only the unit prices, so that the distinct quantities $\Lambda_i$ are the inverse of the physical return of the land concerned: this is the required land area in order to produce a unit quantity of wheat, which therefore varies from one land to another according to its fertility.

This system of equations is interesting, first of all because it makes it possible to calculate without difficulty the amount of the different rents $\rho_i$, all the other variables being known. It is also because the total rent is necessarily nil (it is recalled that the absence of rent was a necessary condition for calculating prices, which are average prices). This implies that the following equation is always satisfied: $\sum \rho_i \Lambda_i = 0$

But it immediately follows that there are lands where rents are necessarily negative, and that this is the *sine qua non* condition for the simultaneous existence of positive rents, since the sum of the two kinds of rents must necessarily cancel out each other. Thus, land can be classified according to the level of its rent, ranging from the one where the positive rent is the
highest, noted $\rho_1$, to that where the rent is the most negative, noted $\rho_n$, through a land where
the rent is null, for instance for $i = g$ (with consequently $\rho_g = 0$). We have $\displaystyle\sum_{i=1}^{i=g} \rho_i \ge 0$, and we
have also necessarily $\displaystyle\sum_{i=g+1}^{i=n} \rho_i < 0$: the sum of the positive rents is equal to the sum of the
negative rents. It should be emphasized that the order of land fertility is deduced from the
order of rents, but that the land marked $\Lambda_1$ is not necessarily the one where the physical yield
(in quantity of wheat per hectare) is the highest. In other words, $\Lambda_1$ is not necessarily the
smallest area, and the property $\Lambda 1 < \Lambda 2 < \dots < \Lambda g < \dots < \Lambda n-1 < \Lambda n$ is not necessarily
verified. This is because in each equation we find also the expressions $\bar{w}_i + a_i p_1 (1 + r_w)$,
which vary with distribution.

One might think that such a system is absurd because it is devoid of all reality: it is hard to see
how landlords could accept to pay a negative rent in order to rent their land! But we will show
through two additional and different examples that this system is not absurd in all cases,
because everything depends on the nature of land ownership.

The first example corresponds to the case where all of the capitalist enterprises of the branch
(the "farmers") have redeemed the land they exploit to the landowners and sell all their wheat
at the price $p_w$. In this case, everything happens as if the best performing firms in the sector
had a rate of profit higher than the average rate of profit, and thus realized an over-profit, and
as if the least performing firms, with higher production costs, obtained a rate of profit lower
than the average rate of profit, realized by the average "farmer" or firm for which the rent is
zero. In fact, we are in a situation where there is no rent, strictly speaking, but because there
are no "pure" landowners. The system, however, can work, since all firms nevertheless make
a profit (otherwise they would not exist), and because those whose profits are reduced
(through negative rents) by the lesser fertility of the lands that they own necessarily bought
these lands at prices lower than that of other lands of greater fertility. The reverse is also true:
the super-profits offset the higher prices paid for the purchase of these other lands.

## 1.2. The price of wheat with a zero total rent and equalization by the State

This situation corresponds to a second example, in which all lands are nationalized, and
consequently owned by the State, which leases them to "farmers", or capitalist enterprises
which use them for the production of wheat. Assuming that the State wishes to maintain the
price of wheat at the $p_w$ level corresponding to the average cost of production, it is sufficient
for this purpose, first, to forego the benefit of an overall net income for the lands it owns, and
then to put in place an equalization scheme. In this case, the State will tax all the companies
for which the rent is positive (which make an additional profit over the average profit rate),
and pay a subsidy to all the companies for which the rent is negative, taxes $(-\rho_1, \dots, -\rho_{g-1})$
and subsidies $(+\rho_{g+1}, \dots, +\rho_n)$ being equal and opposite to the amount of each rent (the
amounts are seen from the point of view of firms, which explains the minus sign). The total amount of taxes is equal to the total amount of subsidies. The net cost to the State is therefore nil, but although it is the owner of the land, it pays nothing to itself. The system of equations

then becomes, with $\rho_i > 0$, $\rho_i < 0$, and $\rho_g = 0$:
$$ \sum_{i=1}^{g-1} \rho_i > 0, \quad \sum_{i=g+1}^{n} \rho_i < 0 $$

$$ \begin{aligned} \bar{w}_1 + a_1 p_1 (1 + r_w) + (\rho_1 - \rho_2) \Lambda_1 + \rho_1 \Lambda_2 &= p_w \\ \bar{w}_2 + a_2 p_1 (1 + r_w) + (\rho_2 - \rho_1) \Lambda_1 + \rho_2 \Lambda_2 &= p_w \end{aligned} $$

(System 2)

$$ \begin{aligned} \bar{w}_{l_g} + a_g p_1 (1 + r_w) + (\rho_g - \rho_{n-1}) \Lambda_{n-1} + \rho_g \Lambda_{n-1} &= p_w \\ \bar{w}_{l_{n-1}} + a_{n-1} p_1 (1 + r_w) + (\rho_{n-1} - \rho_{n-2}) \Lambda_{n-1} + \rho_{n-1} \Lambda_{n-1} &= p_w \\ \bar{w}_{l_n} + a_n p_1 (1 + r_w) + (\rho_n - \rho_{n-1}) \Lambda_{n-1} + \rho_n \Lambda_{n-1} &= p_w \end{aligned} $$

It can be seen that for each category of land the tax or subsidy cancels the amount of rent, this amount being positive when $\rho_i > 0$, or negative when $\rho_i < 0$. Rent is therefore no longer an element of price $p_w$, while these taxes and subsidies create for the State a corresponding positive or negative income, noted in bold characters ($\rho_1\Lambda_1, \rho_2\Lambda_2, \dots, \rho_g\Lambda_g, \dots, \rho_{n-1}\Lambda_{n-1}, \rho_n\Lambda_n$), which replaces rent as an element of the price, knowing that both global rent and global State income are zero. Therefore the price of wheat $p_w$ remains unchanged. Equalization has resulted in a mere transfer of positive or negative incomes from enterprises, for which there is no longer any rent, to the State, whose total income is zero, as is total rent. Thus all firms are equal as regards the rate of profit that they can realize.

These two examples where the price of wheat remains the average price $p_w$ and where global rent is nil clearly show that the previous system is not absurd and can exist under certain conditions. Moreover, there are cases where equalization schemes of this type can be managed by firms themselves, without necessarily involving the State through taxes and subsidies.

### 1.3. The price of wheat with a zero differential rent on marginal land

In the real world, if the State is not the owner of any land, then lands are the property of landowners who lease them to capitalist enterprises. It is clear, first of all, that there can be no negative rent for any landowner, because it would mean that they would have to pay for their land to be used! This implies, first of all, that price $p_w$ is fixed at a level $p'_w$ which totally suppresses the negative rents which appeared in the previous price system, instead of letting these rents appear on the least fertile lands before being compensated by the State.
This price $p'_w$ must be such that it cancels the highest negative rent $\rho_n$ on the least fertile land, noted $\Lambda_n$. This implies that the price $p_w$ be increased by an amount $-\rho_n\Lambda_n$, equivalent to its absolute value $|\rho_n|\Lambda_n$, in order to cancel the negative rent on the marginal land. Since the price of wheat is the same regardless of the category of land, this increase must apply to all lands. The price system must therefore become:

$$ \bar{w}l_1 + a_1p_1(1+r_w) + \rho_1\Lambda_1 + |\rho_n|\Lambda_n = p'_w $$

$$ \text{thus } p'_w = p_w + \rho_i\Lambda_i + |\rho_n|\Lambda_n $$

$$ \bar{w}l_2 + a_2p_1(1+r_w) + \rho_2\Lambda_2 + |\rho_n|\Lambda_n = p'_w $$

(we recall that $\sum_{i=1}^{g-1} \rho_i > 0$ and $\sum_{i=g+1}^{i=n} \rho_i < 0$)

$$ \bar{w}l_g + a_g p_1 (1 + r_w) + \rho_g \Lambda_g + |\rho_n| \Lambda_n = p'_w $$

$$ \text{with } \rho_g = 0 $$

(System 3)

$$ \bar{w}l_{n-1} + a_{n-1}p_1(1+r_w) + \rho_{n-1}\Lambda_{n-1} + |\rho_n|\Lambda_n = p'_w $$

$$ \bar{w}l_n + a_n p_1 (1 + r_w) + (\rho_n - \rho_n) \Lambda_n = p'_w \Rightarrow \bar{w}l_n + a_n p_1 (1 + r_w) + 0 = p'_w $$

For the marginal land at the price level $p'_w$, with $p'_w = p_w + |\rho_n|\Lambda_n$, the rent is therefore nil. On the other hand, firms using land $\Lambda_{n-1}$ will pay a rent equal to $\rho_{n-1}\Lambda_{n-1} + |\rho_n|\Lambda_n$, and so on: firms using land $\Lambda_1$ will pay a rent $\rho_1\Lambda_1 + |\rho_n|\Lambda_n$.

### 1.4. The price of wheat with a zero differential rent and a positive absolute rent on marginal land

The fact that the rent on the least fertile land is zero in price system 3 shows, however, that this case does not yet correspond to a situation where land is appropriated by landowners, which implies that the owner of the least fertile land refuses to provide it to a farmer for free and therefore without any rent. The landowner in question must therefore receive a rent, which corresponds to what Marx calls the absolute rent. Let us call $\rho_a$ the level of this rent per area unit. This rent can be collected only if it is passed on in price, and since there is only one price of wheat, therefore all the landowners will benefit from this rent, at the same level for all. The price system then becomes: $p_w'' = p_w' + \rho_a\Lambda_i$

$$ \bar{w}l_1 + a_1p_1(1+r_w) + (\rho_1+\rho_a)\Lambda_1 + |\rho_n|\Lambda_n = p_w'' $$

$$ \text{we have } p_w'' = p_w' + \rho_a\Lambda_i $$

$$ \bar{w}l_2 + a_2p_1(1+r_w) + (\rho_2+\rho_a)\Lambda_2 + |\rho_n|\Lambda_n = p_w'' $$

$$ \text{with } \rho_a > 0 \text{ and } \rho_n < 0 $$

(System 4)

$$ \bar{w}l_g + a_g p_1 (1 + r_w) + (\rho_g + \rho_a) \Lambda_g + |\rho_n| \Lambda_n = p_g'' \text{, with } \rho_g = 0 $$