• the subsequent maintenance of these various elements, which is likely to prolong or shorten their total life, depending on the quality and intensity of this maintenance;

• the pace of technical progress, which can render obsolescent and therefore lead to the decommissioning of certain elements of this fixed capital, even if their maximum possible lifetime is still far from being reached.

The only way to avoid these insurmountable problems is therefore to measure the stock of fixed capital excluding depreciation, and thus to its original value, as long as it is actually used in the production process. This is what is usually called valuation at historical values or prices.

In any case, there is one last reason for doing so, and this is the criterion of practice. Indeed, the practice of capitalist enterprises when it comes to estimating the profitability of an anticipated investment is as everyone knows to use always for this purpose as the cost of capital (of the investment in question) the initial actual cost, in order to calculate what in the jargon of financiers is called ROI ("return on investment"), by equalizing this cost with all the future incomes expected from this investment. Similarly, when it comes to comparing this initial forecast, which led to the realization of a particular investment, to the actual results obtained from the same investment, and therefore to an ex-post evaluation of the merits of this investment, it is obvious that we must reuse the same original value as that used for the initial calculations, that is to say this historical value. This is the method used by financiers and management controllers. Otherwise the comparison between expectations and achievements would no longer make sense.

However, we must be aware that doing so, i.e. using historical values and prices, leads to profit rate levels that will necessarily - and definitely, be lower than using the method taking into account the depreciation of fixed capital. This amounts to increasing the value of capital in proportion to the increase in its lifespan, i.e. by a factor $\frac{t}{\frac{t-1}{2}} = \frac{2t}{t-1}$ (see equations 4 and 6).

We can derive the consequence on the rate of profit, using Marx's formulation, which adds variable capital at the denominator. As a simplified example, let us assume that the rate of surplus-value $k$ is 100 %, meaning that $k = \frac{S}{V} = 1$ or $S = V$.

Then, for an amount of surplus-value that is unchanged at the numerator of the expression of this rate, in the simple model without capitalist consumption, this implies that $L_I = L_{II}$, and that the rate of profit calculated with the assumption of depreciation of capital will be:

$$r_I = \frac{L_I}{L_I T + L_I} = \frac{L_I}{L_I \frac{t-1}{2} + L_I} = \frac{2}{t+1} \text{ instead of:}$$
$$r_{II} = \frac{L_I}{L_I t + L_I} = \frac{1}{t+1}.$$

For the rate of profit calculated with the value of capital based on the assumption of its depreciation, this method corresponds to a multiplication by a factor 2: the profit rate is doubled.

If, on the other hand, the current definition of the rate of profit were used, considering it as the simple ratio of profit to fixed capital only, the resulting rate of profit, calculated with the

assumption of depreciation, would be multiplied by a factor $\frac{\frac{1}{t-1}}{\frac{2}{t}} = \frac{2t}{t-1}$.

In order to give a numerical example, let us assume that the average lifespan of fixed capital is 10 years. Then with Marx's definition the profit rate would go up from 9.1 % without depreciation of fixed capital to 18.2 % with depreciation. Using the current definition without variable capital in the denominator, the profit rate would go from 10 % to 22.2 %.

We had voluntarily left aside the question of the rate of profit when we had explored simple reproduction in the simplified model of chapter 11, where that was no capitalist consumption. Having dealt with the measurement of the stock of capital, we can now go further in analyzing the rate of profit and its evolution in connection with the organic composition of capital, since we have all the elements to determine this rate and its evolution. Like in the present section, this will be done first in the context of this simple model.

### 3. The average rate of profit in the simple model

In order to simplify the presentation, which is the least that we can expect from a simplified model, we will deal with only one case as regards parameter $u$, i.e. the share of the value of available fixed capital obtained at each stage of exchanges taking place within section I, among capitalists of this section. To make sure that this case is not artificial, and as close as possible to reality, it seems better to stick to the assumption of case n° 2 examined above in chapter 11, where parameter $u$ is equal to $b$, i.e. to $1-a$. This amounts to considering that capitalists of section I (apart from those selling fixed capital to section II capitalists) have a homogeneous behavior, and fix their price with the same margin rate $1-u=1-b=a$.

The reason for this choice comes from the fact that in developed economies coefficient $a$, as the share of the value of fixed capital employed in the sector producing this same fixed capital (therefore quite close to the share of employment in this sector), seems to have a magnitude ranging between 0.2 and 0.3 (or between 20 and 30%). Moreover with this assumption we also arrive at a margin rate (the same coefficient $a$, by construction) which is consistent with the values found in these economies, where they are in the same area. If, on the contrary, we
had adopted the hypothesis corresponding to case n° 1, with u equal to a, the margin rate 1-a would have been between 70 and 80%, which would have been rather unrealistic.

Having adopted this assumption, we now have almost all of the elements that make it possible to determine the evolution of the rate of profit in the framework of the model, as depending on the variation of the organic composition of the capital, designated by symbol q, whose determination remains to be clarified. This is the reason why we will begin by this determination.

The value of the capital stock has been determined in the last section, with the assumption that the average lifespan of fixed capital, which is a given technical parameter, is noted t. Adding the simplifying assumption that t is the same in both sections, we can deduce the organic composition of capital, both in terms of values and in terms of prices.

## 3.1. The organic composition of capital

### 3.1.1. The organic composition of capital in terms of values

#### 1) At global level

Let us recall that $S = L_I$, $V = L_{II}$ and that according to equation (4) we have $K = L_I t$

This gives us:

$$q = \frac{K}{V} = \frac{L_I t}{L_{II}} \qquad (9)$$

We know that $k = \frac{L_I}{L_{II}}$ is the rate of surplus-value, and that consequently $L_I = kL_{II}$

This implies that in terms of values $q = \frac{kL_{II}}{L_{II}} t = kt$, or conversely that $k = \frac{q}{t}$ (10)

#### 2) In section II (for the sake of simplicity t has the same value here as in section I)

The organic composition of capital in section II is expressed by ratio $q_{II} = \frac{K_{II}}{V_{II}}$ (11)

We know (see the first row of table 5 in chapter 11) that the value of fixed capital invested annually in section II is:

$$(1-a)L_I = (1-a)kL_{II} \qquad (12)$$

It follows that the value of the total accumulated fixed capital is:

$$(1-a)L_I t = (1-a)kL_{II} \qquad (13)$$
Knowing moreover (see equation (4) in chapter 11) that the value of labor power in section II is:

$V_{II} = \frac{L_{II}}{1+k}$, we obtain:

$$q_{II}(\text{values}) = \frac{(1-a)ktL_{II}}{\frac{L_{II}}{1+k}} = (1-a)(1+k)kt \quad (14)$$

### 3) In section I

The organic composition of capital in section I is expressed by ratio $q_I = \frac{K_I}{V_I}$

The annual investment in values in section I is $q_I = \frac{K_I}{V_I}$ and the capital stock is therefore $aL_I t$ (assuming that the average lifespan of fixed capital is the same in both sections).

The value of labor power in section I is $\frac{L_I}{1+k}$ (see also p. 214), so that we have:

$$q_I(\text{valeurs}) = \frac{aL_I t}{\frac{L_I}{1+k}} = at(1+k) \quad (15)$$

### 3.1.2. The organic composition of capital in terms of prices

#### 1) At global level

We know that the annual investment in terms of prices is $L_I(\frac{1}{1-a})$. At global level, the price of the capital stock is therefore $K = \frac{tL_I}{1-a}$ and the price of labor-power (or total wages) is $W$, which is equal to $L_I + L_{II}$ (with $\bar{w} = 1$).

The organic composition of capital in terms of prices, or $q$ is therefore:

$$q(\text{prices}) = \frac{K}{W} = \frac{\frac{tL_I}{1-a}}{L_I + L_{II}} = \frac{\frac{tL_I}{1-a}}{L_I + \frac{L_I}{k}} = \frac{\frac{t}{1-a}}{\frac{t}{k}} = \frac{kt}{(1-a)(1+k)} \quad (16)$$

#### 2) In section II
We know that the price of investments during the period in section II is $L_I$. The price of the stock of capital for section II is therefore $K_{II} = L_I t$ and that of labor-power (section II wages) is $W_{II} = L_{II}$ (with $\bar{w}=1$).

It follows that the organic composition of capital in terms of price is:

$$q_{II}(\text{prices}) = \frac{K_{II}}{W_{II}} = \frac{L_I t}{L_{II}} = \frac{L_I t}{\frac{L_I}{k}} = kt \qquad (17)$$

It has therefore the same value as the organic composition in terms of values at global level.

### 3) In section I

We know that the price of investments during the period in section I is $L_I \frac{a}{1-a}$. The price of the stock of capital for section I is therefore $K_I = L_I t \frac{a}{1-a}$ and that of labor-power (section I wages) is $W_I = L_I$ (with $\bar{w}=1$).

It follows that the organic composition of capital in section I, in terms of prices is:

$$q_I(\text{prices}) = \frac{K_I}{W_I} = \frac{\frac{aL_I t}{1-a}}{L_I} = \frac{at}{1-a} \qquad (18)$$

We have now all the parameters that we need to define the average rate of profit.

## 3.2. The average rate of profit

At first sight, the rate of profit is normally supposed to belong to the theoretical universe of prices: in the neo-classical theory it is supposed to be the rate of return of capital as a factor of production, whose components are measured at their price. In the theory of production prices, it is defined as a ratio between a share of the surplus and the means of production made of basic goods, both being measured in prices. However in Marx's formulation it is defined as a ratio of values, which has the interest to make it invariable to prices and fluctuations in prices.

This is the reason why we will start by calculating the overall profit rate of the system keeping for the moment the same formula as that used by Marx, which we recalled in the introduction to this chapter. This implies to make calculations from data in value, at least for the time being, and first at global level. Furthermore, simple reproduction implies that there are different rates in both sections.
### 3.2.1. The average rate of profit at global level in terms of values

On this basis, we therefore have
$$r = \frac{S}{K+V} = \frac{\frac{S}{V}}{\frac{K}{V}+1} = \frac{\frac{L_I}{L_{II}}}{\frac{L_I}{L_{II}}t+1}$$

Or, with the adopted notations,
$$r = \frac{k}{kt+1} \qquad (19)$$

and since $k = \frac{q}{t}$ (see equation (10) above), we get
$$r = \frac{q}{(q+1)t} \qquad (20)$$

In these two last equations, r is a function of only two variables: either the rate of surplus-value k or the organic composition of capital q, and the average lifespan of capital t. Now let us look at the profit rates in section I and section II, because we showed that in our theoretical framework there is no equalization of the rate of profit: let us state again that the difference between these rates is even a necessary condition for realization of the product and thus the reproduction of the system.

### 3.2.2. The average rate of profit in section II

Starting with section II producing consumption goods, we know that the value of labor-power $V_{II}$ is equal to $\frac{L_{II}}{1+k}$, that surplus-value amounts there to $\frac{k}{1+k} L_{II}$, and that accumulation during the period is equal in value to $L_I(1-a)$. In a simple reproduction regime, the capital stock in value is therefore simply $L_I(1-a)t$.

As a result, the profit rate in Section II expressed in terms of values is:

$$r_{II}(\text{values}) = \frac{\frac{k}{1+k} L_{II}}{L_I(1-a)t + \frac{L_{II}}{1+k}} = \frac{kL_{II}}{kL_{II}(1-a)(1+k)t + L_{II}} = \frac{k}{kt(1-a)(1+k)+1} \qquad (21)$$

The profit rate in the same section, expressed now in terms of prices, has a much simpler expression, since its formulation leads to the following result:

$$r_{II}(\text{prices}) = \frac{L_I}{L_I t + L_{II}} = \frac{\frac{L_I}{L_{II}}}{\frac{L_I}{L_{II}}t+1} = \frac{k}{kt+1} \qquad (22)$$

Thus, the profit rate in the consumption goods section, expressed in the price system, is absolutely identical to the overall rate of profit, or the average rate of profit at global level,
expressed in the value system, as shown in equation (19). We can also observe that the rates of profit calculated in terms of values and in terms of prices coincide in this same section II only when:

$(1-a)(1+k)=1$, from equations (19) and (21)

Which implies that $(1-a) = \frac{1}{1+k}$ or also $1+k = \frac{1}{1-a}$ (23)

This gives us $a = 1 - \frac{1}{1+k} = \frac{k}{1+k}$ or also $k = \frac{1}{1-a} - 1 = \frac{a}{1-a}$ (24)

### 3.2.3. The average rate of profit in section I

Continuing with section I producing fixed capital, we know that the value of labor-power in this section is $\frac{L_I}{1+k}$, and that surplus-value there is therefore $\frac{k}{1+k} L_I$. We know also that fixed capital accumulated during one period in this section represents $a$ times the total value $L_I$ of the fixed capital it produces, and that the total accumulated capital represents $t$ times this value, namely:

$$K_I = aL_I t \qquad (25)$$

So the rate of profit in the section producing fixed capital (let us call it $r_I$), expressed in terms of values, is:

$$r_I(\text{values}) = \frac{\frac{k}{1+k} L_I}{aL_I t + \frac{L_I}{1+k}} = \frac{kL_I}{(1+k)aL_I t + L_I} = \frac{k}{a(1+k)t+1} = \frac{k}{akt+at+1} \qquad (26)$$

As for the rate of profit of the same section I producing fixed capital, but this time in the price system, it is equal to:

$$r_I(\text{prix}) = \frac{L_I \frac{a}{1-a}}{Lt \frac{a}{1-a} + L_I} = \frac{aL_I}{atLI + (1-a)L_I} = \frac{a}{at+1-a} \qquad (27)$$

We can observe that in section I the rate of profit calculated in terms of values and that calculated in terms of prices coincide only when:

$$\frac{k}{akt + at + 1} = \frac{a}{at + 1 - a}, \text{ which implies that:}$$

$$a^2 tk + a^2 t + a = atk - ak + k \qquad (28)$$
By putting $k$ in factor, it comes: $k(-a^2t + at - a + 1) = a^2t + a$

That gives then by rearranging: $k[-(-a^2t - a) + (at + 1)] = a(at + 1)$

From which we get: $k(at + 1)(1 - a) = a(at + 1)$

$$ \text{This gives us } k = \frac{a}{1-a}, \text{ which is equivalent, as we already saw in (24) to } a = \frac{k}{1+k} \quad (29) $$

This shows that the condition of equality of rates of profit expressed in value and in price in section I is exactly the same as that already determined for section II (see equation 24).

### 3.2.4. Equalization of average profit rates in value and price

To conclude on the average profit rate in the simple model, it has already been determined in the value system (see equation (19)), where it is equal to:

$$ r(values) = \frac{k}{kt + 1} $$

We still have to determine the average rate of profit in the price system, where it is equal to:

$$ r(prices) = \frac{L_I \frac{1}{1-a}}{L_I \frac{1}{1-a} t + (L_I + L_{II})} = \frac{L_I \frac{1}{1-a}}{\frac{L_I kt}{(1-a)k} + \frac{(1-a)L_I(1+k)}{(1-a)k}} = \frac{k}{kt + (1-a)(1+k)} \quad (30) $$

It can be deduced that the overall rate of profit expressed in terms of values and that expressed in terms of prices can be identical provided that we have:

$$ \frac{k}{kt + 1} = \frac{k}{kt + (1-a)(1+k)}, \text{ which simply implies that the following relationship be verified:} $$

$$ (1-a)(1+k) = 1 \Leftrightarrow (1-a) = \frac{1}{1+k} \Leftrightarrow 1+k = \frac{1}{1-a} $$

This expression is equivalent to: $a = 1 - \frac{1}{1+k} = \frac{k}{1+k}$, and also to $k = \frac{1}{1-a} - 1 = \frac{a}{1-a}$

$$ \text{Therefore we may write : } a = \frac{k}{1+k} \Leftrightarrow k = \frac{a}{1-a} \quad (31) $$

This is exactly the same condition as that required for the rate of profit calculated in terms of values and that calculated in terms of prices to coincide in section II producing consumption goods, which resulted from the equation (24), as well as in section I producing fixed capital (see equation 29). For this reason, from now on we will call these relations given by equations
(30) between the magnitudes of *a* and *k*, and which ensure an identical rate of profit in values and in prices *the proportion of Marx*.

At the end of all these calculations, it seemed useful to gather in a summary table the different variables and parameters of the simplified model, in terms of values as well as in terms of prices. They all appear in Appendix 3 at the end of the book, where they are provided in table 1 of this appendix. In this table the rate of profit *r* is expressed as a function of *k*, the rate of surplus-value, rather than as a function of *q*, the organic composition of capital, since most of the variables are expressed in terms of *k*. But we could easily switch from one variable to the other, given that according to equation (10) we have $q=kt$, or conversely that $k = \frac{q}{t}$.

All these preliminary definitions were necessary before exploring what they mean in terms of the evolution of the profit rate within the framework of the simplified model without capitalist consumption. This evolution is the object of the next chapter.

## References

Marx, Karl (1894). *Capital: A Critique of Political Economy. Volume III: The Process of Capitalist Production as a Whole*. Edited by Friedrich Engels. Source: Institute of Marxist-Leninism, USSR, 1959. Marx/Engels Archive [online]. Available at: https://www.marxists.org/archive/marx/works/download/pdf/Capital-Volume-III.pdf
Accessed 29 Aug 2018

Tsoulfidis, Lefteris (2006). *Falling Rate of Profit and Overaccumulation in Marx and Keynes. Political Economy Quarterly*, Issue 43 (N°3), pp. 65-75.
# Chapter 19. The evolution of the rate of profit as defined by Marx in the simple model

We will continue in this new chapter to work on the basis of the Marxist definition of the rate of profit which puts variable capital, or the value of labor power, at the denominator of the expression giving this rate, since it is a necessary condition if we want to express it as a function of both the rate of surplus-value, i.e. $k = \frac{S}{V}$, and the organic composition of capital, defined as $q = \frac{K}{V}$. It is on this basis that its evolution will be analyzed and graphically represented, both in terms of values and in terms of prices. A numerical example corresponding to the simplified model will also be provided.

## 1. The evolution of the rate of profit and the proportion of Marx

At this stage, and as we just saw that the overall rate of profit expressed on the basis of values and that expressed on the basis of prices differ in the general case, one must inevitably ask what is the right formulation of this rate to be used with a view to discussing its evolution.

Certainly the rate that corresponds to the empirical reality is the rate expressed on the basis of prices. But as the point of departure of the last chapter was to check the validity of the Marxist law of the tendency of the rate of profit to fall, it nevertheless seems preferable to start, at least temporarily, with the Marxist definition of the rate of profit, expressed consequently on the basis of values. Let us recall that at global level and in terms of values we have:

$$r = \frac{k}{kt + 1}, \text{ equivalent to } r = \frac{q}{(q+1)t}$$

The Marxist definition is indeed that which must be retained if we wish to check the validity (or absence of validity) of Marx's law on the basis of his own hypotheses, knowing that this formula remains valid for the rate of profit expressed in terms of prices if the proportion of Marx is respected (see equations (30) at the end of last chapter).

Moreover, we have seen that the overall rate of profit expressed on the basis of values is in any case equivalent to the rate expressed on the basis of prices in section II producing consumption goods. But it will be shown anyway below, in the course of this chapter, that the results obtained on the basis of values do not differ fundamentally from those obtained by using the general formula of the rate of profit expressed in terms of prices.

The formula defining the rate of profit on the basis of values implies first of all that it is obviously impossible to consider the organic composition of capital $q = kt$ as independent of the rate of surplus-value $k$. On the contrary $q$ depends directly on this rate, since it is defined and deduced from it by the application of a $t$ coefficient that could be called the coefficient of immobilization of fixed capital, itself reflecting the lifespan of this capital. If this lifespan is
assumed to be constant, this observation refers to the fact that the organic composition of capital cannot vary without a prior variation in the ratio between the value of fixed capital accumulated in each period and the value of labor power. This supposes a concomitant variation of the rate of surplus-value, and consequently of the proportion according to which workers are distributed between the two sections of production – with $\bar{w}$ being the same in both sections (taking into account the remark made on this subject above).

But the formula expressing the rate of profit allows in a second step to analyze the way in which the evolution of the organic composition of capital acts on the evolution of this rate, and this is where the interest of such a formula turns out to be the greatest.

To analyze the evolution of the rate of profit, we will continue to assume that the average lifespan of fixed capital is constant, which corresponds well to a situation of simple reproduction, and anyway does not seem to be an assumption too remote from reality, at least in the short-term or even the medium-term, given the inertia normally displayed by a variable like the value of the accumulated capital stock.

On this basis, it appears that function $r = f(k) = g(q)$ is a homographic function, i.e. a well-known category of functions, that can be represented as a quotient of two affine functions, of

the form $f(x) = \frac{ax+b}{cx+d}$. The derivative of such a function is $f'(x) = \frac{\begin{vmatrix} a & b \\ c & d \end{vmatrix}}{(cx+d)^2}$.

This function $r = g(q)$ can easily be rewritten in such a way that we get:

$$r = \frac{q}{(q+1)t} = \frac{1}{t}\left(1 - \frac{1}{q+1}\right) = \frac{1}{t}[1 - f(q)] \quad (1)$$

Where $\frac{1}{t}$ with $t$ as a constant is mathematically a scaling factor, which therefore does not affect the direction of variation of the function.

$$\text{Moreover, it is trivial that function } f(q) = \frac{1}{q+1} \text{ is decreasing on } [0; +\infty[ \quad (2)$$

From that we deduce that the initial function $r = f(k) = g(q)$ is increasing on $[0; +\infty[$.

This means that any increase in the organic composition of capital corresponds to an increase in the rate of profit, and conversely that any decrease in the organic composition of capital corresponds to a fall in the rate of profit. This leads us to a result that is strictly opposite to that of Marx and the Marxist tradition! This means that at least within the framework of this simplified model, which is in any case consistent with Marx's approach - apart from the transfer of value of fixed capital, the law of the tendency of the rate of profit to fall cannot be sustained on mathematical grounds, and is therefore not logically valid. On the contrary, it must be replaced by the law of the tendency of the rate of profit to rise! To be sure, this is a
fundamental finding, even though this phenomenon is based on quite restrictive assumptions, and must therefore be relativized.

As stated at the beginning of this section (on the previous page), this observation remains true if we replace the formula on the basis of values used above to express *r* by the formula in terms of prices (see equation (30) at the end of last chapter).

$$ \text{We have indeed: } r(\text{prices}) = \frac{k}{kt+(1-a)(1+k)} = \frac{q}{[q+(1-a)(1+k)]t} = \frac{1}{t} \left( \frac{q}{q+\alpha} \right) \quad (3) $$

$$ \text{Considering temporarily that } \alpha = (1-a)(1+k) $$

Then, since *t* can again be considered as a scaling factor, we can write:

$$ r = \frac{1}{t} f(q) = \frac{1}{t} \left( \frac{q}{q+\alpha} \right) \quad \text{which gives us} \quad f'(q) = \frac{\alpha}{(q+\alpha)^2} \quad (4) $$

We deduce that $f'(q)$ has the same sign as $\alpha$ at the numerator, and therefore that:

$$ \alpha = (1-a)(1+k) > 0 \Leftrightarrow f(q) = \frac{1}{q+\alpha} \quad \text{is strictly increasing} \quad (5) $$

$$ \alpha = (1-a)(1+k) < 0 \Leftrightarrow f(q) = \frac{1}{q+\alpha} \quad \text{is strictly decreasing} \quad (6) $$

Now, we know that *k* is by definition always positive, and that *a* fortiori $k > -1$. So the variations of *f*(*q*) depend only on the sign of $(1-a)$. And since we also know that we always have $a < 1$ (otherwise this would imply that the whole fixed capital of the productive system is used in section I), it follows that the rate of profit $r = f(q)$ expressed in terms of prices is always a strictly increasing function of the organic composition of capital *q* and the rate of surplus-value *k* since $q = kt$ (and *q* is deduced from *k*).

It has thus been demonstrated, to be sure within the framework of the simplified model, but on the basis of Marx's own hypotheses - apart from the transmission of value of fixed capital - that in this particular case the law of the tendency of the rate of profit to fall must be replaced by the law of the tendency of the rate of profit to rise, and this even in the context of a definition of the rate of profit in terms of prices, which is absent in Marx.

This tendency to rise, however, has a limit, since the analysis of function $r = f(k) = g(q)$ shows that the rate of profit tends asymptotically towards a maximum which is equal to $r = \frac{1}{t}$, when the organic composition of capital tends to infinity: the maximum rate of profit is then equal to the inverse of the average lifespan of fixed capital. This is an interesting result, since conversely there does not seem to be a limit to the fall in the profit rate in the Marxist analysis, when the organic composition increases while the rate of surplus-value is supposed
to remain constant (these last two evolutions being incompatible simultaneously within the framework of this model). But this new law of the tendency of the rate of profit to rise will perhaps be more meaningful to the reader by means of a graphical representation, which is given below.

## 2. A graphical representation of function $r = f(q)$ in terms of values

Indeed, we observe in figure 19.1 that function $r = f(q)$ is increasing according to the organic composition of capital, and that we therefore have, for a constant fixed capital lifespan, an increase in the rate of profit, linked to the increase in the organic composition of capital, and which tends towards an asymptote equal to the inverse of the lifespan of fixed capital. But we see also that the whole curve representing function $r = f(q)$ moves downwards when the lifespan of fixed capital gradually increases, which is another factor of increase of the organic composition of capital, but which thus tends in the case of this variable to provoke a decline in the rate of profit.

In passing, this shows that the rate of profit is a complex variable, which, even at the level of simplification where we are here, depends, like the organic composition of capital, of two variables, the rate of surplus-value and the lifespan of fixed capital. Admittedly, in a situation of simple reproduction where the system is supposed to reproduce identically from one period to the next, the lifespan of fixed capital is not supposed to change. It is thus the variation of the rate of surplus-value only which provokes that of the organic composition of capital.

<figure><img src="image_4.png" /><figcaption>Figure 19.1 - Representation of function $r (values) = f(q) = q / (q + 1) t$, for different values of $t$</figcaption></figure>
We will see in the next chapter that even in a situation of simple reproduction, other variables, related to the introduction of the consumption of capitalists, will come into play. Moreover, the real world is not in a situation of simple reproduction. Taking expanded reproduction into account could only lead to the introduction of an even greater number of variables, and this is why it will be tackled only briefly in this book, and by means of numerical examples.

The fact remains that the rate of profit in the real world is thus the result of a large number of parameters (among which tax rules regarding capital depreciation have already been mentioned). Their interaction can only lead to permanent fluctuations in the average rate of profit, upwards or downwards, just as the diversity of production conditions from one sector to another and one firm to the other can only play in the direction of a differentiation of profit rates between sectors and firms.

This is the reason why, although the present part started from a question about the relevance of a law that concerns the rate of profit, the reflection carried out here leads once more to put into perspective the importance of this law, especially with regard to the rate of profit, or rather the rates of profit, which are recorded ex-post.

On the other hand, at microeconomic level and in the context of the preparation of investment decisions, it is clear that the expected rate of profit, or what is often referred to as the projected internal rate of return (IRR), certainly continues to play an important and useful role. Here we find the notion of marginal efficiency of capital dear to Keynes, which corresponds to the expected rate of profit as a key variable in the decision to invest. But then we are at microeconomic level, because each firm has its own appreciation of what the level of marginal efficiency of capital should be, whether it leads to a given investment or not. Again, this observation goes into the direction of a multiplicity of profit rates which itself leads to relativizing very strongly the importance of such a parameter as the average rate of profit, both as an expected rate and as a realized rate.

It seemed useful nevertheless to make more concrete this new law of the tendency of the rate of profit to rise with the help of a numerical example, which is thus developed hereafter.

### 3. A numerical example corresponding to the simplified model

This example uses the static comparative method. Two systems A and B are considered in a state of simple reproduction for which organic composition differs, and consequently the distribution of the produced value between each of the two sections of the productive system. In addition, it is assumed that the average lifespan of fixed capital in both cases is 7 years, meaning that with our notations $t = 7$. The values for the two systems A and B are:

Section I Section II

A $L_I = 100$ $L_{II} = 400$, so that the organic composition is $q_A = \frac{K_A}{V_A} = \frac{100 \times 7}{400} = 1,75$
$$B \quad L_I = 200 \quad L_{II} = 400, \text{ so that the organic composition is } q_B = \frac{K_B}{V_B} = \frac{200 \times 7}{400} = 3,5$$

We can see that the rate of surplus-value increases sharply from A to B and goes from $k_A = 0.25 = 25\%$ to $k_B = 0.50 = 50\%$. The value of the total accumulated capital increases from $K_A = 700$ for system A to $K_B = 1400$ for system B. We thus obtain:

$$r_A = \frac{S_A}{K_A + V_A} = \frac{100}{700 + 400} = 9,1\%, \text{ or } r_A = \frac{k_A}{k_A t + 1} = \frac{0.25}{0.25 \times 7 + 1} = \frac{0.25}{2.75} = 0.091$$

$$r_B = \frac{S_B}{K_B + V_B} = \frac{200}{1400 + 400} = 11,1\% \text{ or } r_B = \frac{k_B}{k_B t + 1} = \frac{0.50}{0.50 \times 7 + 1} = \frac{0.50}{4.5} = 0.111$$

We can observe that the rise in the organic composition of capital, which doubles (like the rate of surplus-value) from 1.75 to 3.5, is accompanied not by a fall, but by a rise in the profit rate from 9.1% to 11.1%.

It is also interesting to note in passing that this increase in the rate of profit is parallel to a decline in the ratio $\frac{Y}{K}$, which to some extent can be considered as representative of the "return" on capital, and which goes from:

$$\frac{100 + 400}{700} = 0,71 \quad \text{to} \quad \frac{200 + 400}{1400} = 0,43$$

The reader may, however, wonder whether the seemingly paradoxical result thus achieved is not due to the restrictive nature of the model assumptions, namely the lack of capitalist consumption and a situation of simple reproduction. If we actually introduce into the model the consumption of capitalists, we will see that the situation will indeed get more complicated and that the results will change. This will be shown in the chapters to come.

## 4. The evolution of the rate of profit in terms of prices

We saw in section 2 of chapter 17, starting from a situation where the rate of surplus-value was $k = \frac{L_I}{L_{II}}$, that an attempt to increase profits in section II had ultimately resulted in a decline in employment and value produced, and this in each of the two sections of the productive system, from $L_I$ to $L'_I$ and from $L_{II}$ to $L'_{II}$, and thus to a new rate of surplus value $k'' = \frac{L'_{I}}{L'_{II}}$. At the same time, it was found that the amount of profits had decreased in each of the two sections: from $L_I$ to $L'_I$ in section II, and from $L_I \frac{a}{1-a}$ to $L'_{I} \frac{a}{1-a}$ in section I.
What about the evolution of the rate of profit, globally and in each of the two sections, using now its expression in terms of prices ?

To answer this question, let us begin by emphasizing that the new rate of surplus-value k" may be either greater than or lower than the initial rate of surplus-value k, depending on whether the rate of decline of L<sub>I</sub> is lower or higher than the rate of decline of L<sub>II</sub>.

To appreciate the evolution of the overall rate of profit in terms of prices as a function of the evolution of the rate of surplus-value *k*, let us recall that its expression is:

$$
r(\text{prices}) = \frac{k}{kt + (1-a)(1+k)}.
$$

We know that in this simple model without capitalist consumption it is an increasing function of *q*. It is likewise an increasing function of *k*, since we know that $q = \frac{kt}{(1-a)(1+k)}$, in terms of prices (see equation 16 in chapter 18). Figure 19.2 corresponding to $r(prices) = f(k)$ is given for different values of *t* and for *a* = 0.25.

<figure><img src="image_5.png" /><figcaption>**Figure 19.2 - Representation of function r (prices) = f (k) = k / (kt + (1-a)(1+k)) with a = 0.25**</figcaption></figure>

What happens in the case of an upward trend in the rate of surplus-value? In this simple model it implies that in section I $L_I$, corresponding to value produced (and by extension to employment) increases at a higher rate or decreases at a slower rate than $L_{II}$, the value produced and by extension the employment in section II. The graph above shows that an
increase in the rate of surplus-value *k* on the x-axis leads to an increase in the overall rate of profit on the y-axis.

This can happen even in the case of a decrease in the absolute amount of profits, if this
decrease takes place at a lower rate than a simultaneous decrease in the value of labor power.
But this rise of the global rate of profit can correspond to a divergent evolution for the rate of
profit of each section:

- The rate of profit will increase at the same time in section II, where we have also
$r_{II} = \frac{k}{kt+1}$, since it is similarly an increasing function of *k* (its derivative is $\frac{1}{(kt+1)^2}$).

- Whereas in section I, where we have $r_I = \frac{a}{at+1-a}$, the rate of profit does not depend on the rate of surplus value k, but depends on the evolution of parameter a, i.e. the share of the value of fixed capital goods going to section I.

Figure 19.3 reflects this evolution.

<figure><img src="image_6.png" /><figcaption>**Figure 19.3 - Representation of function r_I (prices) = f (a) = $\frac{a}{at+1-a}$**</figcaption></figure>

On the x-axis in figure 19.3 we have no more the rate of surplus-value but parameter *a*
instead. It can thus be seen that an increase in the rate of surplus-value *k* can be accompanied
by both an increase in the overall rate of profit and an increase in the profit rate in section II,
but with a change in the rate of profit in section I which remains independent of this evolution
of the rate of surplus value. Indeed it can take place in any direction - because it depends only
on the evolution of parameter *a*: if *a* decreases then the rate of profit *r*<sub>I</sub> will itself fall. Its
evolution can therefore diverge from that of *r* and *r*<sub>II</sub>. But all this can be accompanied by a
simultaneous decline in the amount of profits in each of the two sections. This kind of possible evolution of the system goes clearly in the direction of its structural instability.

The least that can be concluded from these contrasting developments is that competition between capitalists, insofar as it seems to act first in the direction of an equalization of the monetary wage rate and/or the level of employment, does not act in the direction of equalizing profit rates. This equalization thus appears to be nothing else than a myth of political economy, to which Marx himself succumbed in Capital, but which has no serious theoretical justification or empirical consistency.

## 5. Rate of profit, rate of accumulation and intensity of capital

To conclude this chapter, it may be interesting briefly to review the relationship between the rate of profit, the rate of accumulation and the intensity of capital in this simplified model.

To this end, we will define here the rate accumulation as the ratio of on the one hand the value of the gross increase in the stock of capital in a period (corresponding to the surplus-value which is reinvested rather than consumed), over the overall gross product of the period, on the other hand. This is different from the usual Marxist definition, which at the numerator takes the net increase in the stock of capital in a period, since we must take into account the fact that in a situation of simple reproduction this net increase is obviously zero. In fact this rate of accumulation is similar *ceteris paribus* to what is usually called by national accountants the investment rate (appearing also in business statistics), i.e. the ratio of gross tangible investment to value added, in terms of prices.

As for capital intensity, it is defined here as the amount of fixed *capital* present in relation to labor. At the level of the production process, it may be estimated by the capital to labor ratio.

We know from equation (33) in chapter 11, concerning the simple model where the whole amount of surplus-value is invested, that the investment ratio in terms of values is:

$$ \frac{I}{Y} = \frac{L_I}{L_I + L_{II}} = \frac{kL_{II}}{L_{II}(k+1)} = \frac{k}{k+1} \quad (7) $$

It is clear that this investment rate or rate of accumulation is an increasing function of *k*, the rate of surplus-value.

As for the ratio of capital intensity, it is by definition, also in terms of values:

$$ \frac{K}{L_I + L_{II}} = \frac{L_I t}{L_I + L_{II}} = \frac{kt}{k+1} \quad (8) $$

Capital intensity is also an increasing function of *k*.

Not surprisingly, we can observe that in this model the ratio of capital intensity is equal to *t* times the rate of investment, *t* being the average lifespan of fixed capital.
We know also that in this simple model the global rate of profit $r = \frac{k}{kt+1}$ is an increasing function of the rate of surplus-value $k$, since the derivative of its function is positive. In fact it is immediate that it is the same, for an identical reason, for both the rate of accumulation and the rate of capital intensity: both rates increase with the rate of surplus-value.

Going one step further it is interesting to compare the evolution of the profit rate and that of these two last rates, which can be done quite easily by simply writing the expressions giving the ratios of the rate of profit over these two rates.

Indeed the ratio of the rate of profit over the rate of investment is :

$$ \frac{r}{\frac{I}{Y}} = \frac{\frac{k}{kt+1}}{\frac{k}{k+1}} = \frac{k+1}{kt+1} \qquad (9) $$

And the ratio of the rate of profit over the rate of capital intensity is:

$$ \frac{r}{\frac{K}{L_I + L_{II}}} = \frac{\frac{k}{kt+1}}{\frac{k}{k+1}} = \frac{k+1}{(kt+1)t} \qquad (10) $$

The derivatives of both functions are:

$$ f'\left(\frac{r}{\frac{I}{Y}}\right) = \frac{1-t}{(kt+1)^2}, \text{ and :} \qquad (11) $$

$$ f'\left(\frac{r}{\frac{K}{L_I + L_{II}}}\right) = \frac{t-t^2}{[(kt+1)t]^2} \qquad (12) $$

Since the average lifespan of fixed capital $t$ is always $> 1$, it is obvious that both derivatives are negatives, which means that both ratios are a decreasing function of the rate of surplus-value $k$.

What all these equations tell us in economic terms is simply that in a situation of simple reproduction where all surplus-value is invested and transformed into fixed capital:

1) an increase in the rate of surplus-value $k$ will necessarily induce an increase in the rate of profit, as well as an increase in both the rate of investment and the rate of capital intensity;
2) the increase in the rate of profit will take place at a slower pace than the increase in the rates of investment and of capital intensity.

At the end of this chapter, and referring to the interrogation that we had at its beginning concerning the validity of the Marxian law of the tendency of the rate of profit to fall in response to an increase in the organic composition of capital, we can now state a principle which goes against this law:

**Principle 41:** In a situation of simple reproduction where there is no capitalist consumption and all surplus-value is invested, the rate of profit rises with the increase in organic composition of capital, both being the result of an increase in the rate of surplus-value.
# Chapter 20. The rate of profit as defined by Marx and its evolution in the model with capitalist consumption

When the model with capitalist consumption was initially defined, the question of the rate of profit in this model was left aside, and we have now to define it. Beforehand, we need first to define the stock of fixed capital and the organic composition of capital in this model. This last variable is indeed needed if we continue to define the rate of profit as Marx does, i.e. precisely as a function of the organic composition of capital. We will then be able to understand its evolution, precisely as a function of this organic composition. This will be analyzed both in terms of values and in terms of prices. We will show finally that the validity of the Marxist law is dependent on the share of capitalist consumption in total consumption.

## 1. Capital stock and the organic composition of capital

### 1.1. Capital stock

#### 1.1.1. Capital stock in terms of values

In this model, as we saw previously, surplus-value is increased by an amount corresponding to capitalist consumption, and is now equivalent to $S_c = L_I + cL_{II}$. However in terms of values the fact that only a part of this surplus-value is now invested does not change the fact that this part is still equivalent to the value of fixed capital produced in a period, which remains $L_I$. Therefore there is no reason for the measurement of the stock of capital to change as compared to the simple model.

Thus at global level this stock stays as:

$$K = L_I t \tag{1}$$

In section I and section II this stock is by definition:

$$K_I = aL_I t \tag{2}$$

$$K_{II} = (1-a)L_I t \tag{3}$$

#### 1.1.2. Capital stock in terms of prices

Although in the Marxist definition the analysis is carried out in terms of values, it seems worth it to use this opportunity to give also the expression of the capital stock in terms of prices. At global level we saw in chapter 12 that from equation (39) and (40), and with $\alpha = \frac{a}{1-a}$, the price of fixed capital, the product of section I was:
$$I = \frac{L_I}{(1-a)(1-c^*)} \quad \text{or also} \quad I = L_I(1+\alpha+\gamma^*+\alpha\gamma^*) \tag{4}$$

This implies that with an average lifespan of $t$ years the price of the stock of capital is:

$$K = \frac{L_I t}{(1-a)(1-c^*)} \quad \text{or} \quad K = L_I t (1+\alpha+\gamma^*+\alpha\gamma^*) \tag{5}$$

In section II producing consumption goods, we can deduce similarly the price of the stock of fixed capital, which we call $K_{II}$, from the price of annual investment $I_{II}$, given by equation (37) in chapter 12:

$$K_{II} = \frac{(1-a)L_I t}{(1-a)(1-c^*)} = \frac{L_I t}{(1-c^*)}, \quad \text{which gives us, setting } \frac{c^*}{1-c^*} = \gamma^*:$$

$$K_{II} = \frac{L_I t}{(1-c^*)} = L_I \left( \frac{c^*}{1-c^*} + 1 \right) t = L_I t (1+\gamma^*) \tag{6}$$

Following the same procedure for section I producing fixed capital we get the price of the stock of fixed capital from that of its annual investment given by equation (38), which gives:

$$K_I = \frac{aL_I t}{(1-a)(1-c^*)} = \frac{a}{1-a} L_I t \frac{1}{1-c^*}, \quad \text{which gives us, again with } \frac{a}{1-a} = \alpha$$

$$K_I = \alpha L_I (1+\gamma^*) t = L_I t (\alpha + \alpha\gamma^*) \tag{7}$$

From equations (6) and (7) we can check that the total stock of capital can also be written, as the sum of $K_I$ and $K_{II}$, and that we find the same result as that given by equation (4):

$$K = K_I + K_{II} = L_I t (\alpha + \alpha\gamma^*) + L_I t (1+\gamma^*) = L_I t (1+\alpha+\gamma^*+\alpha\gamma^*)$$

## 1.2. The organic composition of capital

### 1.2.1. The organic composition of capital in terms of values

At global level the organic composition of capital is obviously modified, compared to the simple model, because the value of labor power $V$ is reduced by capitalist consumption and becomes $V_c$, so that $q$ becomes $q_c$ (we recall that $q = \frac{K}{V} = \frac{L_I t}{L_{II}} = kt$)

$$q_c(\text{values}) = \frac{K}{V_c} = \frac{L_I t}{L_{II}(1-c)} = \frac{kt}{1-c} = \frac{q}{1-c} \tag{8}$$
$$ \text{From which one can obtain similarly: } \frac{q_c}{t} = \frac{k}{1-c} \Rightarrow q_c = \frac{kt}{1-c} \qquad (9) $$

Since we know that by definition $k_c = \frac{k+c}{1-c}$, we can deduct from this equation that:

$$ k_c = \frac{k+c}{1-c} = \frac{k}{1-c} + \frac{c}{1-c} \Rightarrow k_c = \frac{q_c}{t} + \frac{c}{1-c} \qquad (10) $$

$$ \text{Remembering that we had } \frac{c}{1-c} = \gamma, \text{ we can also write: } k_c = \frac{q_c}{t} + \gamma \qquad (11) $$

$$ \text{And since equation (10) above implies that: } \frac{k}{1-c} = k_c - \frac{c}{1-c} = k_c - \gamma \qquad (12) $$

$$ \text{From equation (11) we check that we have again } q_c = (k_c - \gamma)t = \left[k_c - \frac{c}{1-c}\right]t = \frac{kt}{1-c} \qquad (13) $$

One can also calculate the organic composition of capital in the system of values for each section. In section I, the organic composition of capital becomes:

$$ q_{Ic}(\text{values}) = \frac{atL_t}{\frac{L_t}{1+k_c}} = at(1+k_c) = at\frac{1+k}{1-c} \qquad (14) $$

The same calculation for the organic composition of capital in section II results in the following expression:

$$ q_{IIc}(\text{values}) = \frac{(1-a)tL_t}{\frac{L_t}{1+k_c}} = \frac{(1-a)tL_t}{L_t} = kt(1-a)(1+k_c) = kt(1-a)\frac{1+k}{1-c} = \frac{kt(1-a)(1+k)}{1-c} $$

From equation (9) we know that $q_c = \frac{kt}{1-c}$. We can thus replace $\frac{kt}{1-c}$ by $q_c$ in this last equation, which gives us:

$$ q_{IIc}(\text{values}) = kt(1-a)\frac{1+k}{1-c} = q_c(1-a)(1+k) \qquad (15) $$

### 1.2.2. The organic composition of capital in terms of prices

Equation (4) in subsection 1.1.2. above gave us the price of fixed capital produced in section I. As regards the organic composition of capital in terms of prices, and taking as usual $\bar{w}=1$, it is then:

$$ \text{- For the overall organic composition, i.e. } q_c(\text{prices}) = \frac{K_c(\text{prices})}{V_c(\text{prices})} = \frac{K_c(\text{prices})}{W(\text{prices})} $$
$$q_c(prices) = \frac{L_I t(1+\alpha+\gamma^*+\alpha\gamma^*)}{L_I + L_{II}} = \frac{L_I t(1+\alpha+\gamma^*+\alpha\gamma^*)}{L_I \left(1+\frac{1}{k}\right)} = \frac{kt(1+\alpha+\gamma^*+\alpha\gamma^*)}{(1+k)(1-a)(1-c^*)} \quad (16)$$

But we know from equation (4) that $(1+\alpha+\gamma^*+\alpha\gamma^*) = \left(\frac{1}{1-a} \cdot \frac{1}{1-c^*}\right)$

$$\text{So that we can write also: } q_c(prices) = \frac{kt}{(1+k)(1-a)(1-c^*)} \quad (17)$$

- For the organic composition in terms of prices in section I, we know $K_{Ic}$ from equation (7) above in subsection 1.1.2.:

$$q_{Ic}(prices) = \frac{K_{Ic}(prices)}{W_{Ic}(prices)} = \frac{\frac{atL_I}{(1-a)(1-c^*)}}{L_I} = \frac{at}{(1-a)(1-c^*)} = at(1+\alpha+\gamma^*+\alpha\gamma^*) \quad (18)$$

- For the organic composition in terms of prices in section II, we also know $K_{IIc}$ from equation (6) above in subsection 1.1.2.:

$$q_{IIc}(prices) = \frac{K_{IIc}(prices)}{W_{IIc}(prices)} = \frac{\frac{tL_I}{1-c^*}}{L_{II}} = \frac{\frac{tL_I}{1-c^*}}{\frac{L_I}{k}} = \frac{kt}{1-c^*} = kt(1+\gamma^*) \quad (19)$$

We have defined now all the variables that are needed in order to express the rate of profit in this model of simple reproduction with capitalist consumption.

## 2. The average rate of profit as defined by Marx in the model with capitalist consumption

As we already pointed out, the rate of profit is defined by Marx in terms of values, but can be expressed as a function of various parameters. In this section we will calculate its expression first as a function of the rate of surplus-value, and second as a function of the organic composition of capital.

### 2.1. The average rate of profit expressed as a function of the rate of surplus-value $k_c$

By expressing $r_c$ as a function of the rate of surplus-value $k_c$, the formula giving the average rate of profit $r_c$ at global level in the model with capitalist consumption thus becomes:
$$r_c(\text{values}) = \frac{\frac{S_c}{V_c}}{\frac{K_c}{V_c} + 1} = \frac{\frac{L_I + cL_{II}}{L_{II}(1-c)}}{\frac{L_I}{L_{II}(1-c)}t + 1} = \frac{\frac{L_I}{L_{II}(1-c)} + \frac{c}{1-c}}{\frac{L_I}{L_{II}(1-c)}t + 1} = \frac{\frac{k+c}{1-c}}{\frac{kt}{1-c} + 1} = \frac{k_c}{q_c + 1} \quad (20)$$

Using equation (10) above in subsection 1.2.1., from which we can derive that:

$$\frac{q_c}{t} = k_c - \frac{c}{1-c} \Rightarrow q_c = \left[ k_c - \frac{c}{1-c} \right] t, \text{ we can therefore write:}$$

$$r_c = \frac{k_c}{q_c + 1} = \frac{k_c}{\left[ k_c - \frac{c}{1-c} \right] t + 1} \quad (21)$$

And since $\frac{c}{1-c} = \gamma$, then we can define the rate of profit at global level $r_c$ as a function of the rate of surplus-value $k_c$ only, which was the objective of this sub-section. Indeed it can now be written as:

$$r_c = \frac{k_c}{(k_c - \gamma)t + 1} \quad (\text{with } k_c > \gamma) \quad (22)$$

## 2.2. The average rate of profit expressed as a function of the organic composition of capital $q_c$

### 2.2.1. The average rate of profit at global level

In order to express $r_c$ as a function of the organic composition of capital $q_c$, we can use for this purpose equation (11): $k_c = \frac{q_c}{t} + \gamma$. Then the formula giving the average rate of profit in the model with capitalist consumption becomes:

$$r_c(\text{values}) = \frac{\frac{q_c}{t} + \frac{c}{1-c}}{q_c + 1} = \frac{q_c + \frac{c}{1-c}t}{(q_c + 1)t} = \frac{q_c + \gamma t}{(q_c + 1)t} \quad (23)$$

### 2.2.2. The average rate of profit in section I

As regards the expression in the value system of the rate of profit in section I, in order to obtain it we just need to remember first that the rate of surplus-value is the same in both sections and in the whole economic system (we always have $k_c = k_{Ic} = k_{IIc}$), and to go back to the formula defining it, namely:
$$r_{Ic}(\text{values}) = \frac{\frac{S_{Ic}}{V_{Ic}}}{\frac{K_I}{V_{Ic}} + 1} = \frac{k_{Ic}}{q_{Ic} + 1} = \frac{k_c}{q_{Ic} + 1} = \frac{\frac{q_c}{t} + \gamma}{q_{Ic} + 1} = \frac{q_c + \gamma t}{(q_{Ic} + 1)t} \quad (24)$$

This expression can also be written in terms of $k$, i.e. the primary rate of surplus-value (the rate defined in the simple model), since all these variables are already known:

$$r_{Ic}(\text{values}) = \frac{k_c}{at(1+k_c)+1} = \frac{\frac{k+c}{1-c}}{at\frac{1+k}{1-c}+1} = \frac{k+c}{at(1+k)+1-c} \quad (25)$$

### 2.2.3. The average rate of profit in section II

Lastly, as regards the expression of the profit rate in Section II, in order to obtain it in the system of values, it suffices to go first to the formula defining it, namely:

$$r_{IIc}(\text{values}) = \frac{k_{IIc}}{q_{IIc} + 1} = \frac{k_c}{q_{IIc} + 1} = \frac{\frac{q_c}{t} + \gamma}{q_{IIc} + 1} = \frac{q_c + \gamma t}{(q_{IIc} + 1)t} \quad (26)$$

Then we can use equation (15) giving the organic composition of capital in section II, which, like for the rate of profit in section I, allows that this expression be written also in terms of $k$, the primary rate of surplus-value, which gives us:

$$r_{IIc}(\text{values}) = \frac{k_c}{kt(1-a)(1+k_c)+1} = \frac{\frac{k+c}{1-c}}{kt(1-a)\frac{1+k}{1-c}+1} = \frac{k+c}{kt(1-a)(1+k)+1-c} \quad (27)$$

We note that this average rate of profit in section II can thus be defined in terms of the variables of both the simple model: $k$, and the model with capitalist consumption: $k_c$ and $c$.

### 2.2.4. The condition for the equalization of profit rates in both sections

Incidentally, we can ask ourselves what are the conditions for the profit rates $r_{Ic}$ and $r_{IIc}$, defined in terms of values, to be identical in each of the two sections. Since the two numerators in the two equations giving $r_{Ic}$ and $r_{IIc}$ are identical and equal to $k+c$ (see equations (25) and (27), to obtain $r_{Ic} = r_{IIc}$ we just need that the denominators of the two ratios expressing these profit rates in each of the two sections be equal, namely that:

$$at(1+k_c)+1-c = kt(1-a)(1+k_c)+1-c$$
This identity can be very easily simplified, just by removing $(1-c)$ and dividing each term on both sides by $t(1+k_c)$, which leaves us with the following identity:

$$a=k(1-a) \text{ or } k=\frac{a}{1-a} \text{ which is equivalent to } a=\frac{k}{1+k} \qquad (28)$$

We thus note that despite the modified definition of the rate of profit when the consumption of capitalists is taken into account, the verification of this equality between the rates of profit in section I and section II, in terms of values, implies always the same condition as that which was needed for the equalization of the rates of profits in terms of values and in terms of prices in the simple model, meaning that the structure of the system must be consistent with what we called the proportion of Marx.

## 2.3. The average rate of profit expressed as a function of the variables of the simple model

Indeed it is also possible to express $r_c$ as a function of the organic composition of capital and the rate of surplus-value which were defined in the initial simple model (in the absence of capitalists consumption), thus keeping the notations already used in this simple model, with $k$ and $q$, respectively, for the rate of surplus-value and the organic composition of capital.

Let us recall first that by definition $k_c = \frac{k}{1-c} + \frac{c}{1-c} = \frac{k}{1-c} + \gamma$

$$\text{And since } k = \frac{q}{t} \text{ then } k_c = \frac{q}{(1-c)t} + \gamma \qquad (29)$$

$$\text{Let us recall also that on the basis of equation (8): } q_c = \frac{q}{1-c} = \frac{kt}{1-c} \qquad (30)$$

To express $r_c$ as a function of $k$, we can go back to equation (20), in which we replace in the denominator $q_c$ by its value given by equation (29), and the number 1 by $\frac{1-c}{1-c}$, before simplifying the whole expression by $(1-c)$. We then obtain:

$$r_c(\text{values}) = \frac{k_c}{q_c + 1} = \frac{\frac{k+c}{1-c}}{\frac{kt}{1-c} + \frac{1-c}{1-c}} = \frac{k+c}{kt+1-c} = \frac{k+c}{q+1-c} \qquad (31)$$

To express $r_c$ only as a function of $q$, the organic composition of capital, we just need to start from the same equation (20), and knowing that $q = \frac{L_I t}{L_{II}}$, this makes it possible to obtain:
$$r_c(\text{values}) = \frac{\frac{L_I}{L_{II}} + c}{\frac{L_I}{L_{II}}t + (1-c)} = \frac{\frac{L_I}{L_{II}}t + ct}{qt + (1-c)t} = \frac{q + ct}{(q+1-c)t} \quad (32)$$

### 3. The evolution of the rate of profit with the Marxist definition

Now that the rate of profit has been fully defined within the framework of Marx's formulation, i.e. in terms of values, what can we say about its evolution as a function of the organic composition of capital, again in this same model with capitalist consumption?

#### 3.1. The evolution of the rate of profits at global level in terms of $q_c$

Let us first start from equation (23) above, that is from the expression of the rate of profit $r_c$ as a function of $q_c$, i.e. $r_c = g(q_c) = \frac{q_c + \gamma t}{(q_c + 1)t}$, where $\frac{1}{t}$ can be considered mathematically as a scale factor, which therefore does not affect the variation of this function.

$$\text{So } g(q_c) \text{ can be replaced by the homographic function } \tilde{g}(q_c) = t \cdot r_c = \frac{q_c + \gamma t}{q_c + 1} \quad (33)$$

$$\text{From which we obtain the derivative } \tilde{g}'(q_c) = \frac{1 - \gamma t}{(q_c + 1)^2} \quad (34)$$

We deduce from this last equation that $\tilde{g}'(q_c)$ has the sign of $1-\gamma t$ and therefore, with regard to function $g(q_c)$ which differs only by a scale factor, that we have:

$$g(q_c) \text{ strictly increasing} \Leftrightarrow \gamma t < 1 \quad (35)$$

$$g(q_c) \text{ strictly decreasing} \Leftrightarrow \gamma t > 1 \quad (36)$$

From which we deduce, by expressing $\gamma$ as a function of $c$ (with $0 < c < 1$), that:

$$g(q_c) \text{ is strictly increasing if and only if } \frac{c}{1-c}t < 1,$$

Which implies, since $1-c > 0$, that $ct < 1-c$, or $c(1+t) < 1$

$$\text{So that the rate of profit } r_c = g(q_c) \text{ is strictly increasing if and only if } c < \frac{1}{1+t} \quad (37)$$

$$\text{And } r_c = g(q_c) \text{ is strictly decreasing if and only if } c > \frac{1}{1+t} \quad (38)$$
## 3.2. The evolution of the rate of profit at global level in terms of q

Let us now start from equation (32), i.e. from the expression of the profit rate as a function of $q$, with $r_c(\text{values}) = \frac{q+ct}{(q+1-c)t}$, where $\frac{1}{t}$ can again be considered as a scale factor, which as such does not affect the variation of function $r_c = g(q)$.

In order to examine these variations, let us write: $tr_c = \tilde{g}(q) = \frac{q+ct}{q+1-c}$ (39)

It is easy to get the derivative of $\tilde{g}(q)$, which is $\tilde{g}'(q) = \frac{1-c-ct}{(q+1-c)^2}$. (40)

We deduce from this equation that $\tilde{g}'(q)$ has the sign of $1-c-ct$ and therefore that:

$\tilde{g}'(q) > 0$ implies that $1-c-ct > 0 \Leftrightarrow 1-c(1+t) > 0 \Leftrightarrow c(1+t) < 1$

$\tilde{g}'(q) < 0$ implies that $1-c-ct < 0 \Leftrightarrow 1-c(1+t) < 0 \Leftrightarrow c(1+t) > 1$

Since $g(q)$ differs from $\tilde{g}'(q)$ only by a scale factor, we can infer that:

$r_c = g(q)$ is strictly increasing if and only if $c < \frac{1}{1+t}$ (41)

$r_c = g(q)$ is strictly decreasing if and only if $c > \frac{1}{1+t}$ (42)

It is not surprising to find exactly the conditions set by inequalities (37) and (38) above.

This implies that, unlike what has been demonstrated in the simplified model, the Marxist law of the tendency of the rate of profit to fall in the event of an increase in the organic composition of capital is no longer systematically false when one introduces in the system under review the consumption of capitalists.

It is false only when $c < \frac{1}{1+t}$. On the other hand, it is still valid when $c > \frac{1}{1+t}$.

## 3.3. The graphical representation of the evolution of the profit rate in terms of values

We can first represent the evolution of $r_c$, the rate of profit in terms of values, as a function of the evolution of $q_c$, the organic composition of the capital, on the basis of a given value of the average lifespan of fixed capital, i.e. $t$, which in figure 20.1 is assumed to be 5 years.
<figure><img src="image_7.png" /><figcaption>Figure 20.1 - Representation of function r<sub>c</sub> (values) = g(q<sub>c</sub>) = (q<sub>c</sub> + γt) / (q<sub>c</sub> + 1)t</figcaption></figure>

From this figure 20.1 we can see that there is indeed an increase in the rate of profit corresponding to an increase in the organic composition of capital, for the two values of c (c = 0.10 and c = 0.15) which are lower than c = 0.167 (i.e. c = 1/(1+t) with t = 5 years), as in the simple model without capitalist consumption, which again contradicts, under the condition that c < 1/(1+t), the Marxist law of the tendency of the rate of profit to fall. On the other hand, and contrary to this simple model in which capitalist consumption did not appear, when c exceeds this limit value, which is in fact rather low, the Marxist law of the tendency of the rate of profit to fall becomes true.

It is also quite interesting to note that for this limit value of c, i.e. c = 1/(1+t) the rate of profit remains constant, whatever the value taken by the organic composition of capital. Here the value of this constant is r_c = 0.20, or 20%, which is maintained whatever the level of q_c when c = 0.167. It clearly results from this demonstration that the rate of profit is a function of - among others, the level of capitalist consumption (as a share of total consumption), and as such directly depends on it. This variable thus appears to be a key variable in the functioning of the economic system.

This constitutes indeed quite an interesting finding, since it makes the Marxist law of the tendency of the rate of profit to fall dependent on the level of capitalist consumption, as a share of total consumption, and also indirectly on the average lifespan of fixed capital, the
reason being that this last variable is linked to the level of the total stock of fixed capital,and
is in fact one of its determinants.

On the basis of equation (22), we can also provide a graphic representation of the evolution of $r_c$, the rate of profit, as a function of the evolution of $k_c$, the rate of surplus-value which takes into account the consumption of capitalists, always with the same average lifespan of fixed capital, i.e. with $t = 5$ years. This representation appears in figure 20.2, for five different values of $c$ (0.10, 0.15, 0.167, 0.20, 0.25).

<figure><img src="image_8.png" /><figcaption>Figure 20.2 - Representation of function $r_c$ (values) = $f(k_c) = \frac{k_c}{(k_c - \gamma)t + 1}$ with $k_c > \gamma$</figcaption></figure>

Here also it is easy to see that $c = \frac{1}{1+t}$ constitutes a limit value for which the rate of profit becomes a constant, whatever the level of the rate of surplus-value $k_c$. Above this limit $r_c$ is a decreasing function of $k_c$. We can also observe that when this rate of surplus-value increases, whatever the consumption of capitalists, the rate of profit converges asymptotically towards this same limit, which in the chart above stays at 20 %. It is for instance noticeable that, for a given value of $t$, when the rate of surplus-value reaches 100 %, the differences in the rates of profit deriving from the differences in the values of $c$ are reduced.

3.4. A numerical example for the Marxist model with capitalist consumption

A numerical example will help to better understand how the model works, in two possible
cases, named A and B. Let us take the previous numerical example (see section 3 of chapter
19, p.365-366):
<table><thead><tr><th></th><th>Section I</th><th>Section II</th></tr></thead><tbody><tr><td>A</td><td>L<sub>I</sub> = 100</td><td>L<sub>II</sub> = 200</td></tr><tr><td>B</td><td>L<sub>I</sub> = 200</td><td>L<sub>II</sub> = 200</td></tr></tbody></table>

We now consider two distinct cases:

**1st case:** $c < \frac{1}{1+t}$ and we consider for example that $c = 0,10$ and that $t = 7$

In situation A the organic composition of capital is:

$$q_c = \frac{K}{V_c} = \frac{L_I \cdot t}{L_{II}(1-c)} = \frac{100 \times 7}{200 \cdot (1-0,10)} = \frac{700}{180} = 3,89$$

$$\text{And we also have } \gamma = \frac{0,10}{1-0,10} = \frac{1}{9} = 0.11$$

$$\text{The rate of profit is then } r_c = \frac{q_c + \gamma t}{(q_c + 1)t} = \frac{3,89 + (0,11 \times 10)}{(3,89 + 1) \times 10} = \frac{5,0}{48,9} = 10,2\%$$

In situation B the organic composition of capital has become:

$$q_c = \frac{K}{V_c} = \frac{L_I \cdot t}{L_{II}(1-c)} = \frac{200 \times 7}{200 \cdot (1-0,10)} = \frac{1400}{180} = 7,78 \text{ (so it has doubled)}$$

$$\text{And we still have } \gamma = \frac{1}{9} = 0.11$$

$$\text{The rate of profit is then } r_c = \frac{q_c + \gamma t}{(q_c + 1)t} = \frac{7,78 + (0,11 \times 7)}{(7,78 + 1) \times 7} = \frac{8,56}{61,46} = 13,9\%$$

There is indeed a rise in the average rate of profit resulting from the increase in the organic composition of capital, which in this example has doubled. We are clearly in a case where the Marxist law of the tendency of the rate of profit to fall is not valid.

**2nd case:** $c > \frac{1}{1+t}$ and we consider for example that $c = 0.25$ and that $t = 10$

In situation A the organic composition of capital is:

$$q_c = \frac{K}{V_c} = \frac{L_I \cdot t}{L_{II}(1-c)} = \frac{100 \times 10}{200 \cdot (1-0,25)} = \frac{1000}{150} = 6,67$$

$$\text{And we also have } \gamma = \frac{0,25}{1-0,25} = \frac{1}{3} = 0.33$$
The profit rate is then
$$r_c = \frac{q_c + \gamma t}{(q_c + 1)t} = \frac{6,67 + (0,33 \times 10)}{(6,67 + 1) \times 10} = \frac{10}{76,7} = 13,0\%$$

In situation B the organic composition of capital has become:

$$q_c = \frac{K}{V_c} = \frac{L_t \cdot t}{L_{it}(1-c)} = \frac{200 \times 10}{200 \cdot (1-0,25)} = \frac{2000}{150} = 13,33 \text{ (so it has doubled)}$$

And we always have $\gamma = 0.33$

The rate of profit is then
$$r_c = \frac{q_c + \gamma t}{(q_c + 1)t} = \frac{13,33 + (0,33 \times 10)}{(13,33 + 1) \times 10} = \frac{16,67}{143,3} = 11,6\%$$

In this case, we can see conversely a decrease in the average rate of profit resulting from an increase (doubling) in the organic composition of capital. We are thus here in a situation where the Marxist law of the tendency of the rate of profit to fall is perfectly valid. If we had started this demonstration from the alternative formula, i.e. $r_c = \frac{q+ct}{(q+1-c)t}$ (in terms of $q$ instead of $q_c$), the results that we would have obtained of course would have been strictly identical, in both cases.

## 4. Some preliminary findings about the evolution of the rate of profit defined in terms of values as a function of the organic composition of capital

What the last section has clearly established is that the consumption of capitalists as a share of total consumption, and therefore variable $c$ which measures this share, plays a decisive role in the way other variables, like the organic composition of capital $q_c$ or the rate of surplus-value $k_c$, influence changes in the rate of profit.

The lessons that we can learn from this finding emerge more clearly from an examination of function $f(c) = \frac{1}{1+t}$, of which the representative curve, as we have demonstrated, constitutes a boundary between the two regimes of evolution of the average rate of profit, and which can thus be designated as the invariance frontier of the rate of profit. This is why this curve is shown in figure 20.3.

We note that when the lifespan of fixed capital increases, the share of consumption of capitalists that corresponds to the boundary where the behavior of the rate of profit changes decreases: for $t = 4$ years we have $c = 0.20$, but for $t = 7$ years, we have $c = 0,125$.
The area above the boundary, where we have $c > \frac{1}{1+t}$ corresponds to the situation where the function $r_c = \frac{q_c + \gamma t}{(q_c + 1)t}$ is strictly decreasing and where the rate of profit evolves therefore in the opposite direction of the organic composition of capital. This is the situation which corresponds to a fall in the average rate of profit in the event of an increase in the organic composition of capital, and which therefore validates the Marxist law of the tendency of the rate of profit to fall. It is also the situation where, conversely, the decline in the organic composition of capital is accompanied by an increase in the average rate of profit.

The area below the boundary, where we have $c < \frac{1}{1+t}$, corresponds to the situation where function $r_c = \frac{q_c + \gamma t}{(q_c + 1)t}$ is strictly increasing and where the rate of profit evolves in the same direction as the organic composition of capital. This is a situation opposite to that described by Marx, but which nevertheless already resulted from the operation of the simple model, and which corresponds to an increase in the average rate of profit in the event of an increase in the organic composition of capital. This is also the situation where, conversely, the decline in the organic composition of capital is accompanied by a fall in the average rate of profit.

<figure><img src="image_9.png" /><figcaption>**Figure 20.3 - The invariance frontier of the average rate of profit** $f(c) = \frac{1}{1+t}$</figcaption></figure>

At this stage, it is obviously necessary to ask what is the economic interpretation of this somewhat unexpected mathematical result, or in other words what is the reason for the influence that can be exercised by parameter $c$ on the rate of profit, i.e. by the share of capitalist consumption in total consumption, but also by parameter $t$, the average lifespan of fixed capital. To understand it we must simply return to the basic definition of the rate of profit in Marxist theory.
This rate is defined in the simple model either as: $r = \frac{S}{K+V}$ or as $r = \frac{\frac{S}{V}}{\frac{K}{V}+1}$,

Which corresponds in the model with capitalist consumption to: $r_c = \frac{\frac{S_c}{V_c}}{\frac{K}{V_c}+1}$

It can indeed easily be deduced from this last formula that the influence of c is felt through parameter $V_c$, i.e. the value of labor power, which goes from $V = L_{II}$ in the simple model to $V_c = L_{II} (1-c)$ in the model with capitalist consumption. Indeed the decrease in $V_c$, resulting from a rise of c, makes the numerator increase together with the denominator of the above fraction. But the rate of surplus-value in the numerator necessarily increases less rapidly than the organic composition of capital at the denominator, because the growth of the organic composition of capital is amplified by the intervention of parameter t, since $K = L_I t$ and $q_c = kt$. For low values of c, this mechanism explains that the increase in c corresponds to a decrease in the rate of profit.

However when c continues to rise, there comes a threshold where the increase in surplus-value which also results from the rise of c (since we go from $S = kL_{II}$ to $S_c = (k+c)L_{II}$), which is - in addition the decline in V - another factor that increases the rate of surplus-value at the numerator of the fraction, can no longer offset the rise in the organic composition of capital at the denominator of the fraction. And this threshold is obviously all the lower as the influence of t becomes stronger, which happens with the increase in the lifespan of fixed capital. When this threshold is reached, and if the consumption of capitalists remains at the level corresponding to this threshold, the rate of profit remains constant, whatever the value of the organic composition of capital. Once this threshold is crossed, and it is different for each value of t, the direction of variation of the rate of profit is reversed, which explains why the Marxist law of the tendency of the rate of profit to fall becomes valid again.

However, all these findings have been obtained with a definition of the rate of profit as a ratio of values. To fully analyze the Marxist law, it is therefore interesting to see what happens to this law if we define the rate of profits in terms of prices, because in the real world the rate of profit is calculated as a ratio of prices, and not as a ratio of values. Since the Marxist law links the rate of profit to the organic composition of capital we need nevertheless to keep the price (and not the value) of labor-power at the denominator of the fraction, which is also at the core of this Marxist definition. This is what will be done in the next section.
# 5. The evolution of the rate of profit in terms of prices

In order to define the rate of profit in terms of prices, we need first to determine the price of the capital stock and to define the organic composition of capital in terms of prices, which has already been done in section 1 of this chapter, and therefore makes it easy to define this rate of profits.

## 5.1. The expression of the rate of profit in terms of prices

So far and again in this chapter we started by defining first the rate of profit at global level, before defining it for section I and section II. However to express the rate of profit at global level is quite complicated in terms of prices, as we shall see, which explains that in this subsection we will do the opposite, and go from the simplest formulation to the most complicated one. We will thus start by defining the rate of profit in section I.

### 5.1.1. The rate of profit in section I

To calculate the rate of profit in terms of prices in section I, let us recall that capitalists in this section pay to their workers' wages $W_I$ for an amount $L_I$ (by posing $\bar{w}=1$). They make a profit $\Pi_I$ for an amount $L_I(\alpha + \gamma^* + \alpha\gamma^*) = L_I\pi_I$ which is provided by equation (48) in chapter 12. They use fixed capital resulting from yearly investments whose annual price, given by equation (38), is $I_I = L_I(\alpha + \alpha\gamma^*)$. This corresponds to a total quantity of fixed capital employed equal to $tI_I$. The profit rate in terms of prices in section I follows:

$$r_{cI} (\text{prices}) = \frac{L_I (\alpha + \gamma^* + \alpha\gamma^*)}{tL_I (\alpha + \alpha\gamma^*) + L_I} = \frac{\alpha + \gamma^* + \alpha\gamma^*}{t(\alpha + \alpha\gamma^*) + 1} = \frac{(1+\alpha)(1+\gamma^*) - 1}{t\alpha(1+\gamma^*) + 1} = \frac{1+\alpha - \frac{1}{1+\gamma^*}}{t\alpha + \frac{1}{1+\gamma^*}}$$

Since we know that we have also: $\frac{1}{1+\gamma^*} = \frac{1}{1+\frac{c^*}{1-c^*}} = 1-c^*$, then we can write:

$$r_{cI} (\text{prices}) = \frac{1+\alpha-(1-c^*)}{t\alpha+(1-c^*)} = \frac{\alpha+c^*}{t\alpha+1-c^*} \qquad (43)$$

It is recalled that parameter $\alpha$ gives a synthetic account of the distribution of fixed capital in value between the two sections: when $\alpha$ increases, this means that the share of fixed capital going to section I increases, and vice versa. Moreover we remember that we have already defined $\gamma = \frac{c}{1-c}$, which accounts for the distribution of consumption, expressed in terms of values, between workers and capitalists, $c$ being the capitalists' share. It is also recalled that, as defined in section 2 of chapter 12, $c^*$ is a parameter 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 will thus be $\gamma^* = \frac{c^*}{1-c^*}$. Thus $c^*$ is different from $c_I$, the share of capitalists of section I in total consumption. We have shown in chapter 12 that $\frac{c^*}{1-c^*}$ corresponds to the fraction of the gross profit margin intended for consumption of all capitalists from section I over the value of fixed capital that they sell to section II. This explains that $c^*$ and $c_I$ are linked by a rather complicated relationship, given by equation (28) of chapter 12, and which is: $c_I = \gamma^* \frac{k(1-c)}{(1+k)}$.

### 5.1.2. The rate of profit in section II

To calculate the profit rate in Section II, it is recalled that capitalists of this section pay wages $W_{II}$ for an amount equivalent to $L_{II}$ (with $\bar{w}=1$), make a profit $\Pi_{II} = k_c L_{II}$, and use it partially to buy from those of section I the quantity of fixed capital that they need, of a value $L_I(1-a)$, for a price equal to $I_{II} = \frac{L_I}{1-c^*}$. The price of the total fixed capital in function in section II is $tI_{II}$. The rate of profit in terms of prices in section II then follows (it is recalled that $L_I = kL_{II}$):

$$r_{cII} = \frac{\Pi_{II}}{t \cdot I_{II} + L_{II}} = \frac{k_c L_{II}}{\frac{tkL_{II}}{1-c^*} + L_{II}} = \frac{k_c (1-c^*)}{kt + 1 - c^*}$$

Since we know that $k_c = \frac{k+c}{1-c}$ and $\frac{1}{1-c^*} = 1 + \frac{c^*}{1-c^*} = 1 + \gamma^*$, this can be written:

$$r_{cII} = \frac{k+c}{(1-c)[kt(1+\gamma^*)+1]} \quad (44)$$

### 5.1.3. The overall rate of profit

Lastly, with regard to the overall rate of profit in the price system, its calculation is complicated by the fact that it is the sum of two types of profit derived from the two sections of the productive system. These two types of profit, which follow two different logics, are aggregated in the numerator of the fraction which defines the rate of profit: in section II, profits are constituted in a logic that reflects the levy of surplus-value, through the purchase of consumer goods by workers; in section I, profits are obtained on the occasion of the sales of fixed capital, in a logic of redistribution of this same amount of surplus-value between capitalists of the two sections. The calculation is therefore as follows, with the price of fixed capital given by equation (5) above in this chapter:
$$r_c(\text{prices}) = \frac{\Pi_I + \Pi_{II}}{K + V} = \frac{L_I \left( \frac{a}{1-a} + \frac{c^*}{(1-a)(1-c^*)} \right) + k_c L_{II}}{t \frac{L_I}{(1-a)(1-c^*)} + L_I + L_{II}} \Rightarrow$$

$$r_c(\text{prices}) = \frac{L_I \left( \frac{a}{1-a} + \frac{c^*}{(1-a)(1-c^*)} + \frac{k_c}{k} \right)}{t L_I \left[ \frac{1}{1-c^*} \cdot \frac{1}{1-a} \right] + L_I \frac{1+k}{k}}$$

This gives us:

$$r_c(\text{prices}) = \frac{\alpha + (1+\alpha)\gamma^* + \frac{k_c}{k}}{t \left[ (1+\alpha)(1+\gamma^*) + \frac{1+k}{kt} \right]} = \frac{\alpha + \gamma^* + \alpha\gamma^* + \frac{k_c}{k}}{t \left[ 1 + \alpha + \gamma^* + \alpha\gamma^* + \frac{1+k}{kt} \right]}$$

We know also that $k_c = \frac{k}{1-c} + \frac{c}{1-c} = k \frac{1}{1-c} + \frac{c}{1-c} = k(1+\gamma) + \gamma$, from which we obtain:

$$r_c(\text{prices}) = \frac{\alpha + \gamma^* + \alpha\gamma^* + \frac{k(1+\gamma)+\gamma}{k}}{t \left[ 1 + \alpha + \gamma^* + \alpha\gamma^* + \frac{1+k}{kt} \right]} = \frac{k(\alpha + \gamma^* + \alpha\gamma^*) + k(1+\gamma)+\gamma}{kt(1+\alpha+\gamma^*+\alpha\gamma^*)+1+k}$$

$$rc(\text{prices}) = \frac{k(1+\alpha+\gamma+\gamma^*+\alpha\gamma^*)+\gamma}{kt(1+\alpha+\gamma^*+\alpha\gamma^*)+1+k} \quad (45)$$

In the above equation, the rate of profit in terms of prices is expressed as a function of $k$, i.e. the primary rate of surplus-value. But at this point, since we want still to explore the validity of the Marxist definition even when it is expressed in terms of prices, it is now possible and equally interesting to define the rate of profit as a function of the organic composition of capital. We know indeed, as we saw it above (see equations 10 to 12 in this chapter) that:

$$q = kt, \quad q_c = \frac{q}{1-c} = \frac{kt}{1-c}, \quad \text{and that} \quad k_c = \frac{q_c}{t} + \frac{c}{1-c} = \frac{q_c}{t} + \gamma$$

We can thus rewrite $r_c(\text{prices})$ as a function of the initial organic composition of capital $q$:

$$r_c(\text{prices}) = \frac{\frac{q}{t}(1+\alpha+\gamma+\gamma^*+\alpha\gamma^*)+\gamma}{q\left(\frac{1}{t}+1+\alpha+\gamma^*+\alpha\gamma^*\right)+1}, \quad \text{and since} \quad \frac{1}{1-c} = 1+\gamma, \text{ we obtain:}$$
$$r_c(\text{prices}) = \frac{q(1+\alpha+\gamma+\gamma^*+\alpha\gamma^*)+\gamma t}{qt\left(\frac{1}{t}+1+\alpha+\gamma^*+\alpha\gamma^*\right)+t} = \frac{1}{t} \cdot \frac{q(1+\alpha+\gamma+\gamma^*+\alpha\gamma^*)+\gamma t}{q\left(\frac{1}{t}+1+\alpha+\gamma^*+\alpha\gamma^*\right)+1} \quad (46)$$

These expressions of the rate of profit, in terms of prices, first as a function of the rate of surplus-value, and second as a function of the organic composition of capital, allow us now to see what is its evolution as a function of both variables.

## 5.2. The evolution of the average rate of profit in terms of prices

It can be seen that the expression above in equation (46), which expresses the rate of profit (in terms of prices) as a function of the organic composition of capital, still remains a homographic function. To discuss the evolution of the rate of profit, it is therefore sufficient to analyze this function. To simplify the discussion, let us temporarily assume that:

$$1+\alpha+\gamma+\gamma^*+\alpha\gamma^*=a \quad (47)$$

$$\frac{1}{t}+1+\alpha+\gamma^*+\alpha\gamma^*=b \quad (48)$$

We also assume that $t$ is constant.

Then function $r(\text{prices})$ becomes with these notations:

$$r_c(\text{prices}) = f(q) = \frac{1}{t} \cdot \frac{aq + \gamma t}{bq + 1} \quad (49)$$

And since $\frac{1}{t}$ can be considered as a constant scale factor, we can rewrite this function as:

$$g(q) = \frac{aq + \gamma t}{bq + 1} \quad (50)$$

The increasing or decreasing nature of $r(\text{prices}) = f(q)$ depends on the positive or negative nature of its derivative, to which we can substitute the derivative of $g(q)$, i.e.:

$$g'(q) = \frac{a - b\gamma t}{(bq + 1)^2} \quad (51)$$

Since its denominator is always positive, the derivative has the sign of its numerator. It follows that:

$$a - b\gamma t > 0 \Leftrightarrow r_c(\text{prices}) = f(q) \text{ is strictly increasing}$$

$$a - b\gamma t < 0 \Leftrightarrow r_c(\text{prices}) = f(q) \text{ is strictly decreasing}$$
The condition of positivity of the derivative $g'(q)$ is thus that $a-b\gamma t > 0$, which gives us,
developing from the definitions of equations (46) and (47) above:

$$1+\alpha+\gamma+\gamma^*+\alpha\gamma^*-(\frac{1}{t}+1+\alpha+\gamma^*+\alpha\gamma^*)\gamma t>0$$

Knowing that $\frac{1}{t}+1+\alpha+\gamma^*+\alpha\gamma^*>0$, because all of its terms are positive, this expression can
be rewritten under the form:

$$\gamma t < \frac{1+\alpha+\gamma+\gamma^*+\alpha\gamma^*}{1+\frac{1}{t}+\alpha+\gamma^*+\alpha\gamma^*}, \text{ which gives } \gamma < \frac{1+\alpha+\gamma+\gamma^*+\alpha\gamma^*}{1+t(1+\alpha+\gamma^*+\alpha\gamma^*)}, \text{ from which we can deduce:}$$

$$\gamma+\gamma t(1+\alpha+\gamma^*+\alpha\gamma^*)<1+\alpha+\gamma+\gamma^*+\alpha\gamma^* \Rightarrow \gamma t(1+\alpha+\gamma^*+\alpha\gamma^*)<1+\alpha+\gamma^*+\alpha\gamma^*$$

From which we draw that $g'(q)>0$ if and only if $\gamma t<1$ \hfill (52)

Conversely $g'(q)<0$ if and only if $\gamma t>1$ \hfill (53)

It is recalled that $\gamma = \frac{c}{1-c}$, and that $c$ is the share of capitalist consumption within total
consumption (with $0<c<1$). By replacing $\gamma$ by its expression as a function of $c$ in inequation
(52), we obtain:

$$\gamma t < 1 \Rightarrow \frac{c}{1-c} t < 1 \Rightarrow ct-(1-c) < 0 \Rightarrow ct+c < 1 \Rightarrow c(1+t) < 1 \Rightarrow c < \frac{1}{1+t}$$

It follows that $f'(q)$ is positive and that function $r (prices) = f(q)$, which determines the
rate of profit as a function of the organic composition of the capital, is increasing when
coefficient $c$ is lower than $\frac{1}{1+t}$, that is when the share of capitalist consumption in total
consumption is less than the inverse of the average lifetime of fixed capital plus one.
Conversely, function decreases when $c$ is greater than $\frac{1}{1+t}$.

To be sure, the calculations that had to be made to achieve this result were rather
cumbersome, if not very complex. In spite of that, we should not be surprised to find again
exactly the same law that was highlighted for the evolution of the rate of profit on the basis of
its initial definition in terms of values, which corresponded to Marx's formulation. The
identity of both results in fact validates these calculations! It is indeed the opposite that would
have been amazing! Profits are indeed the form taken by surplus-value transformed by the
intervention of the price system when it is levied and distributed among capitalists. Thus there
was no reason to imagine that the passage through the price system that makes possible their
actual realization would have resulted in an evolution of the rate of profit that would have
been distinct and therefore a priori different in the value system as compared to the price
system.

We had highlighted in the simplified model which was the subject of chapters 18 and 19 a law
of evolution of the profit rate opposite to and therefore radically different from that of Marx.
But now we see again, this time in terms of prices, that the introduction of capitalist
consumption allows to verify a law inverse to the Marxist one only in the hypothesis where
the share of capitalist consumption in total consumption is lower than a particular threshold,
defined on the basis of the average lifespan of fixed capital.

However, the assumptions that can be made on this average lifespan in the real world show
that this threshold is certainly quite low, and decreases with the extension of this lifespan
(which is a cause of increase in the organic composition of capital). The average lifespan of
fixed capital in capitalist firms in developed countries seems indeed to be in the range of 5
years to 10 years, which corresponds for the share of capitalists consumption in total
consumption, with $c = \frac{1}{1+t}$, to thresholds of 16.7% and 9.1%, respectively. We know also
that the rate of profit decreases when this average lifespan of fixed capital increases, as
figures 1 and 2 have shown (see page 364and 367).

In passing, it should be recalled that the value of fixed assets in operation is reduced when
linear depreciation is introduced, which is equivalent to a shortening of the notional average
lifespan that can be obtained by relating the fixed capital stock at a given time (measured after
depreciation) to the amount of fixed capital invested each year. We have shown above that
$T = \frac{t-1}{2}$. As a result, statistics taking into account the depreciation of fixed capital to
measure its stock as net of depreciation underestimate its amount and therefore its actual
lifespan(for t = 5 years, we thus have T = 2 years). The end-result is that statistical data
relying on such a basis necessarily overestimate the rate of profit.

In any case, as soon as this threshold equal to $c = \frac{1}{1+t}$ is crossed up, that is to say as soon as
the share of capitalist consumption in total consumption becomes is greater than $\frac{1}{1+t}$, for any
given value of t, then the Marxist law of the tendency of the rate of profit to fall due to an
increase in the organic composition of the capital becomes valid. Moreover this increase is not
a mere tendency, but is on the contrary systematic, within the framework of the assumptions
retained. Although his demonstration left much to be desired, Marx's intuition and
conclusions on the subject were therefore largely well founded.

It must be noted, however, that this law is reversible, meaning that in the situation likely to be
the most realistic, or the most plausible, where $c > \frac{1}{1+t}$, in case of a decline - and no longer a
rise, in the organic composition of capital, there is necessarily a rise in the rate of profit. In the
real world this can be the case for economic systems where there is a decline in heavy
industry, and a development of services requiring less fixed capital. This is certainly a circumstance that has not been foreseen by Marx to characterize the evolution of capitalism, but which must not be ruled out.

To conclude temporarily on the question of the evolution of the rate of profit, it is necessary to underline again that all the demonstration carried out so far was on the basis of the Marxist definition of the rate of profit, which constituted the only way to test the validity of Marx's reasoning. But this definition of the rate of profit puts variable capital (the value or price of labor-power) at the denominator of the expression giving this rate of profit. This is not unrelated as we have just seen with the nature of the results obtained, and in fact is indispensable to introduce the organic composition of capital in the picture. But this leads to a definition of the rate of profit that is not the most common. The current definition is indeed simpler and limits itself to put fixed capital, and fixed capital alone, at the denominator of the expression giving the rate of profit.

At the end of this chapter we are now able to recap the system of values and prices in a summary table, similar to table 4 in chapter 11 for the simplified model, but corresponding now to the model with capitalist consumption. We must stress once again that all the content in this table 2, which appears in appendix 3, is based on the Marxist definition of the rate of profit, which has been the only one examined so far, but is not the current definition of the rate of profit, such as that generally used by economic theory. We will examine below in the next chapter what happens if we leave the Marxist definition of the rate of profit to use the current definition only.

Before that the demonstrations that have been made in this chapter allow us nevertheless to formulate a new principle.

**Principle 42:** In a situation of simple reproduction where capitalist consumption is introduced, then the rate of profit defined as Marx does becomes dependent on the share of this capitalist consumption in total consumption. The validity of the Marxist law of the tendency of the rate of profit to fall when the organic composition of capital increases depends therefore on this share. If this share exceeds a threshold linked to the lifespan of fixed hospital and defined by $c = \frac{1}{1+t}$, the Marxist law is valid. If this share is lower than this threshold the Marxist law is not verified and the rate of profit increases on the contrary as a function of the organic composition of capital.
# Chapter 21. The evolution of the rate of profit - based on its current definition - and the share of profits

In this chapter we will see what happens to the rate of profit when it is defined, not on the basis of the Marxist definition, but of the current definition, i.e. without variable capital at the denominator of the fraction giving its value. We will highlight the interest of this current definition, and will show what is the evolution of the rate of profit both in the simple model exposed earlier and in the model with capitalist consumption. This will also be done both in terms of values and in terms of prices. Graphical representations will be provided for all these cases. But we will not confine ourselves only to analyzing the evolution of the rate of profit, and will also address in the second part of the chapter the question of the evolution of the share of profits. We will conclude this chapter by exploring briefly the link between the rate of profit and the share of profits.

## 1. The interest of the current definition of the rate of profit

The ultimate goal of this work is not to develop a pure theory for just the beauty of it, but to provide theoretical tools to begin to shed some light, through the lessons of the models developed so far, on the way the capitalist system actually functions in the real world. This could also help us to better understand the reasons for the long term crisis into which the capitalist system has progressively entered since the late 1970s or early 1980s, in practically all of the most developed countries. This long term crisis is characterized among others by a dramatic surge in inequalities, which has been well described - if not well explained, in Piketty's much acclaimed book "Capital in the 21st Century". I have criticized Piketty's explanations - but not his data and figures - in my article previously cited: "Again on Piketty's *Capital in the Twenty-First Century*".

If one wants to leave the realm of theory to try to understand the evolution of the real world, it is a limitation from which it is advisable to free oneself. In so far as one has to come to grips with the reality of crises, a good point of entry for the analyses conducted so far has indeed been the Marxist theory of crises based on the tendency of the rate of profit to fall, and this was a good reason to base these analyzes on the Marxist definition of the rate of profit. But as we have already pointed out, this definition, by including wages at the denominator of the expression defining the rate of profit, does not correspond to the usual way of defining this variable, which considers it rather as a ratio between profits and the value of fixed capital alone (which includes the value of the intermediate commodities used for its production). Indeed in the real world wages are not advanced one year before the realization of the product.

For this reason it seems preferable to avoid that the reasoning which follows be criticized on the basis of a rejection of the Marxist definition of the rate of profit. This is the reason why this chapter will now be devoted to the determination of the rate of profit, still using the models developed so far, but this time on the basis of the current definition of this rate, which
is also going to simplify its expression. This will allow us to demonstrate that the conclusions drawn so far remain valid within the modified framework of this current definition.

Moreover, and to remain consistent with our questioning about the determining character of the rate of profit, if only because of the demonstration that a multiplicity of profit rates was necessary for the reproduction of the system, this chapter will also examine how a variable that ultimately appears more fundamental is determined in the model: this variable is the share of profits in global income (or output), which corresponds to what is often called the margin rate in its macroeconomic sense.

## 2. The current definition of the rate of profit

Let us recall that this definition only relates surplus-value (in terms of values) or profit (in terms of prices) to fixed capital alone (in value or price).

### 2.1. Within the framework of the simplified model

$$ \text{In terms of values, we have } r(\text{values}) = \frac{S}{K} = \frac{L_I}{L_I t} = \frac{1}{t} \qquad (1) $$

In terms of prices, we know from equation (17) in chapter 11 that total profit is:

$$ \Pi = L_I \left(1 + \frac{a}{u}\right) = L_I + L_I \frac{a}{u} $$

Simplifying again by taking $u=b=1-a$ (as we already did previously), we have:

$$ r_c(\text{prices}) = \frac{\Pi}{K} = \frac{L_I \left(1 + \frac{a}{1-a}\right)}{L_I \frac{1}{1-a} t} = \frac{L_I \frac{1}{1-a}}{L_I \frac{1}{1-a} t} = \frac{1}{t} \qquad (2) $$

We thus find that the rate of profit is the same in its two expressions, both in terms of values and in terms of prices. Moreover with this definition the organic composition of capital does not intervene in the definition of the rate of profit. The latter is therefore independent from the organic composition of capital, and even from the rate of surplus-value $k$, equal to $L_I / L_{II}$.

In fact the rate of profit in the simplified model is only the inverse of the average lifespan of fixed capital. It means that the rate of profit is here purely and simply a technological parameter, knowing however that we are in a situation of simple reproduction, without any capitalist consumption, that the whole surplus-value is invested in fixed capital, and that the rise or fall in the rate of surplus-value, when it occurs, is supposed to have passed on to the totality of the stock of fixed capital (which cannot happen instantaneously, but after a time $t$).
## 2.2. Within the framework of the model with capitalist consumption

### 2.2.1. The rate of profit in terms of values

Let us start first by writing the most simple expression of the rate of profit in terms of values, which is:

$$r_c(\text{values}) = \frac{S_c}{K} = \frac{L_{II}(k+c)}{L_I t} = \frac{L_{II}(k+c)}{k L_{II} t} = \frac{k+c}{kt} \quad (3)$$

From this expression, since we want to check the validity of the Marxist law of the evolution of the rate of profit with its current definition, we can derive a new one expressing $r_c(\text{values})$ in terms of $q_c$, the organic composition of capital.

1) The evolution of the rate of profit in terms of $q_c$, the organic composition of capital

If we want to verify whether the Marxist law of the tendency of the rate of profit to fall with a rise in the organic composition of capital is still valid with this current definition of the rate of profit, it is better to do this checking in terms of values, i.e. in the same way as it was formulated by Marx. First, because it is more simple mathematically, and second because in any case we will see below in subsection 3.5. that the rate of profit defined in terms of prices always evolves like the rate of profit defined in terms of values.

To do that implies to redefine the rate of profit $r_c$ by reintroducing the organic composition of capital $q_c$ in the above equation giving $r_c$. This can be done very simply by just dividing both the numerator and denominator of equation (3) above, i.e. $r_c(\text{values}) = \frac{S_c}{K}$ by the value of labor power, i.e. $V = L_{II}(1-c)$, so that we obtain:

$$r_c(\text{values}) = \frac{S_c}{K} = \frac{\frac{L_{II}(k+c)}{L_I t}}{\frac{L_{II}(1-c)}{L_{II}(1-c)}} = \frac{\frac{k+c}{1-c}}{\frac{kt}{1-c}} = \frac{k_c}{q_c} \quad (4)$$

Obviously we find that the rate of profit $r_c$ is now the ratio of the rate of surplus-value with capitalist consumption, i.e. $k_c$, over the organic composition of capital with capitalist consumption, i.e. $q_c$. This rate $r_c$ is thus clearly a function of two variables: $r_c = f(k_c, q_c)$.

As such this function has no derivative, but a gradient, which we call $G[f(k_c, q_c)]$. The formula giving this gradient is:
$$G[f(k_c, q_c)] = \begin{pmatrix} \frac{\delta f(k_c, q_c)}{\delta k_c} \\ \frac{\delta f(k_c, q_c)}{\delta q_c} \end{pmatrix} = \begin{pmatrix} \frac{1}{q_c} \\ -\frac{k_c}{q_c^2} \end{pmatrix}, \quad \text{which is a field of vectors} \quad (5)$$

This gradient never cancels (which would mean that $q_c$ becomes infinite and that $k_c$ equals zero). Therefore it has no extremum. On the other hand we can see that $\frac{\delta f(k_c, q_c)}{\delta k_c} = \frac{1}{q_c}$ is always positive and therefore that $r_c(\text{values}) = f(k_c, q_c)$ is increasing in terms of $k_c$, for instance on the x axis, whereas $\frac{\delta f(k_c, q_c)}{\delta q_c} = -\frac{k_c}{q_c^2}$ is always negative and therefore is decreasing in terms of $q_c$, for instance on the y axis, this decrease being $-\frac{k_c}{q_c}$ times the increase in terms of $k_c$.

This shows that the Marxist law of the tendency of the rate of profit to fall when the organic composition of capital grows is still verified with the current definition of the rate of profit, and that its validity no longer depends on the value of c like it was the case with the pure Marxist definition, where c had to be higher than $\frac{1}{1+t}$ for the law to be verified (see above subsection 3.2. of chapter 20).

In fact this law can also be highlighted by expressing $r_c = f(k_c, q_c)$ only in terms of $q_c$ and c (or $\gamma$), which is easy to do, remembering that we can write:

$$k_c = \frac{k+c}{1-c} = \frac{k}{1-c} + \frac{c}{1-c} = \frac{kt}{(1-c)t} + \gamma = \frac{q_c}{t} + \gamma, \quad \text{which was equation (11) in chapter 20.}$$

$$\text{Therefore } r_c = g(q_c) = \frac{k_c}{q_c} = \frac{\frac{q_c}{t} + \gamma}{q_c} = \frac{q_c + \gamma t}{q_c t} = \frac{1}{t} \cdot \frac{q_c + \gamma t}{q_c} \quad (6)$$

We can see that $g(q_c)$ is a homographic function of $q_c$, where $\frac{1}{t}$ is a scale factor. From this function we can draw the derivative of $t \cdot r_c = \tilde{g}(q_c)$, at a scale factor close of $g(q_c)$. It is:

$$\tilde{g}'(q_c) = -\frac{\gamma t}{q_c^2} \quad (7)$$
$\tilde{g}'(q_c)$ has the sign of $-\gamma t$, and since both $\gamma$ and $t$ are always positive, $-\gamma t$ is always negative. It follows that, with the current definition of the rate of profit, $r_c = g(q_c)$ is always a decreasing function: an increase in $q_c$ always causes a decrease in the rate of profit $r_c$, for any given value of $\gamma$, and therefore of $c$. However, when $c$ (and thus $\gamma$) are increasing for a given value of $q_c$, then $r_c = g(q_c)$ is going to increase.

2) The evolution of the rate of profit in terms of $k_c$, the rate of surplus-value

We already indicated in the previous page that $\frac{\delta f(k_c, q_c)}{\delta k_c} = \frac{1}{q_c}$ is always positive and therefore that $r_c(\text{values}) = f(k_c, q_c)$ is increasing in terms of $k_c$. In other words, the rate of surplus-value $k_c$ may increase at a rhythm which can be higher than the decrease generated by the increase in $q_c$, the organic composition of capital. From what we saw above, we can deduce that an increase in $r_c$ resulting from an increase in $k_c$ will prevail over a decrease resulting from an increase in $q_c$ if this decrease is less than $-\frac{k_c}{q_c}$ times the increase of $k_c$.

This is the reason why the law of the tendency of the rate of profit to fall when the organic composition of capital rises can only be a tendency, because it can thus be countered by a rise in the rate of surplus-value $k_c$. This rise itself can only come from the rise in its own components, i.e. $k$ and $c$. Let us examine below what results from the evolution of $k$ and $c$.

3) The evolution of the rate of profit in terms of $k$, the primary rate of surplus-value and $c$, the share of capitalist consumption.

We saw above that the most simple expression of the rate of profit, given by equation (3) is:

$$r_c(\text{values}) = \frac{S_c}{K} = \frac{L_{II}(k+c)}{L_I t} = \frac{L_{II}(k+c)}{k L_{II} t} = \frac{k+c}{kt}$$

To begin with, it is clear that $r_c = \frac{k+c}{kt}$ is again a homographic function of $k$, for a given value of $c$ considered as a constant. This makes it possible to describe the evolution of the rate of profit, no longer as a function of the organic composition of capital $q_c$, which does not appear in this expression, but as a function of the rate of "primary" surplus-value $k$ (and not of $k_c$, the rate of surplus-value with capitalist consumption), being noted that $c$ itself appears at the numerator of the expression. This is a simple function, since its derivative is equal to:
$$f'(k) = \frac{-ct}{(kt)^2} \qquad (8)$$

Since we know that $c > 0$ and $t > 0$, it follows that the derivative $f'(k)$ is always negative, and therefore that function $r_c = f(k)$ is always decreasing, which implies that the increase in the primary rate of surplus-value always results in a lower rate of profit, *ceteris paribus*.

But it is easy to show that an increase in $k$ alone does not necessarily translate into a rise in $k_c$ because by definition $k_c = \frac{k+c}{1-c}$, which means that such an increase in $k$ can be countered by a simultaneous decrease in $c$. To show that it can be useful to establish a relation between $k$ and $c$ which will leave $k_c$ as a constant. Let us thus consider that $k_c = h$ with $h$ = constant.

This simply means that the variation of $k$ and $c$ necessarily cancel each other's, in such a way that:

$$\frac{k+c}{1-c} = h \Rightarrow k+c = h(1-c) \Rightarrow k = -c(1+h)+h \qquad (9)$$

Or alternatively: $c = -\frac{k-h}{1+h}$

In other words if $k$ grows, $c$ must decline in such a proportion that equation (9) remains valid, or that for any given value of $k$ there can be a value of $c$ which cancels its influence on $k_c$, and therefore on the rate of profit as a function of $k_c$. To give an example, with $k = 0.25$ and $c = 0.30$ (with $k < 1$ and $c < 1$), we have:

$$k_c = \frac{k+c}{1-c} = 0.7857 \Rightarrow k = -1.7857c + 0.7857$$

Let us take $k_c = h = 0.7857$

Then if $k$ rises to 0.40, then $c$ must be reduced to $c = \frac{k-0.7857}{-1.7857} = \frac{0.4-0.7857}{-1.7857} = 0.216$ for $k_c$ to remain constant. This can easily be checked.

But the fact that the opposite evolution of $k$ and $c$ cancel each other in the expression of $k_c$ does not imply that the same happens in the evolution of $q_c$, the organic composition of capital. Indeed $q_c = \frac{kt}{1-c}$. Thus, taking $t=10$, when $k$ and $c$ go from 0.25 to 0.40 and from 0.30 to 0.216, respectively, then $q_c$ increases from 3.57 to 5.10, which obviously reduces the rate of profit $r_c = f(q_c)$, from 22% to 15.40%. Another way for the organic composition of
capital to grow “autonomously” would be for the average lifespan of fixed capital to increase, which would reduce the rate of profit in due proportions, since $r_c = \frac{k+c}{kt}$ (see equation (3)).

It is clear nevertheless that the organic composition of capital $q_c = \frac{kt}{1-c}$ is itself an increasing function of three variables: i.e. $k$ the primary rate of surplus-value, $c$ the share of capitalist consumption in total consumption (in value), and $t$ the average lifespan of fixed capital. And it is also obvious that the first two variables, $k$ and $c$, if they are growing simultaneously, increase the rate of surplus-value $k_c = \frac{k+c}{1-c}$ through both the numerator and denominator of the equation giving $r_c$, and therefore counterbalance the increase in the organic composition of capital at the denominator of this same equation.

This will maybe appear more easily by providing a second numerical example:

Let us take $t = 10$, $k = 0.25$, and $c = 0.2$

Thus we have:

$$k_c = \frac{0.25+0.20}{0.8} = 0.5625; \quad q_c = \frac{0.25*10}{1-0.2} = 3.125 \quad \text{and} \quad r_c = \frac{k_c}{q_c} = \frac{0.5625}{3.125} = 0.18 = 18\%$$

Then let us imagine that both variables increasing the rate of surplus-value are rising, so that we have now $t = 10$, $k = 0.30$, and $c = 0.3$

$$\text{Therefore } k_c = \frac{0.3+0.3}{0.7} = 0.8571; \quad q_c = \frac{0.30*10}{1-0.3} = 4.2857 \quad \text{and} \quad r_c = \frac{k_c}{q_c} = \frac{0.8571}{4.2857} = 0.20 = 20\%$$

This example shows that an increase in the organic composition of capital $q_c$ resulting from an increase in the primary rate of surplus-value $k$, which alone would induce a decrease in the rate of profit $r_c$, when combined with an increase in the share of capitalist consumption $c$, causes both the rate of surplus-value $k_c$ and the rate of profit $r_c$ to increase, despite the increase in the organic composition of capital which should normally result in the decrease of this rate of profit. This sheds light on the prominent role played by capitalist consumption in the evolution of other key variables of the system, since both the rate of surplus-value and the rate of profit are always an increasing function of this share of capitalist consumption in total consumption.

### 2.2.2. The rate of profit in terms of prices

In terms of prices, the rate of profit can be calculated by referring to its definitions already given for the different sections of the productive system in the previous chapter, with the suppression of the price of labor-power which was at the denominator.
For capitalists of section II, starting from the first expression of equation 44 in chapter 20, and removing the price of labor-power $L_{II}$ (with $\bar{w}=1$) from the denominator, the rate of profit (let us call it $r_{cII}$) is now defined as:

$$r_{cII} = \frac{k_c L_{II}}{ktL_{II}} = \frac{k_c(1-c^*)}{kt} = \frac{(k+c)(1-c^*)}{(1-c)kt} \quad (10)$$

For capitalists of section I, it seems interesting to make a distinction, already introduced previously, between those who produce fixed capital sold to capitalists in section II and those who produce fixed capital sold to capitalists in section I, also going upstream to all other members of this section I (selling intermediate goods to the former).

This separation of capitalists from section I into two sub-categories will provide a possibility to check the coherence of the expression of these rates of profit.

The former make a profit equal to $L_I(a+\gamma^*) = L_I\left(a+\frac{c^*}{1-c^*}\right)$, according to equation (26) in chapter 12, and use a quantity of fixed capital whose price (and not value) is equal to $\frac{L_Iat}{1-c^*}$.

Their rate of profit (let us call it $r'_{cI}$) is therefore defined as:

$$r'_{cI} = \frac{L_I(a+\gamma^*)}{L_Iat(1+\gamma^*)} = \frac{a+\gamma^*}{at(1+\gamma^*)} = \frac{1}{t}\left(1-c^*+\frac{c^*}{a}\right) \quad (11)$$

The second realize a profit equal to $\alpha L_I(a+\gamma^*)$, taken from table 7 in chapter 12, and use an amount of fixed capital whose overall price is $a(\alpha+\alpha\gamma^*)tL_I$. Their rate of profit (let us call it $r''_{cI}$) is therefore:

$$\begin{align*} r''_{cI} &= \frac{\alpha L_I (a + \gamma^*)}{a (\alpha + \alpha \gamma^*) t L_I} = \frac{a + \gamma^*}{(1 - a) (\alpha + \alpha \gamma^*) t} = \frac{1}{t} \left[ \frac{a + \gamma^*}{\alpha (1 - a) (1 + \gamma^*)} \right] \Rightarrow \\ r''_{cI} &= \frac{1}{t} \left( \frac{a + \gamma^*}{a (1 + \gamma^*)} \right) = \frac{1}{t} \left( 1 - c^* + \frac{c^*}{a} \right) \tag{12} \end{align*}$$

This allows to verify that this rate of profit is in fact identical to the previous one, which is logical, under the assumption that parameters $a$ (and therefore $\alpha$) as well as $c^*$ are the same at all levels of the supply chain in section I.

Noting that by definition $a\alpha+a=\alpha$, the average profit rate for section I as a whole is thus:
$$r_{cI} = \frac{L_I(\alpha + \gamma^* + \alpha\gamma^*)}{tL_I(\alpha + \alpha\gamma^*)} = \frac{(1+\alpha)(1+\gamma^*) - 1}{\alpha t(1+\gamma^*)} = \frac{1+\alpha - \frac{1}{1+\gamma^*}}{\alpha t} = \frac{\alpha + c^*}{\alpha t} \quad (13)$$

Similarly, by developing this last equation, we get:

$$r_{cI} = \frac{\alpha + c^*}{\alpha t} = \frac{1}{t} \left( \frac{a + c^* - ac^*}{a} \right) = \frac{1}{t} \left( 1 - c^* + \frac{c^*}{a} \right) \quad (14)$$

For the overall rate of profit, referring to table 2 in appendix 3, we have:

$$r_c(\text{prices}) = \frac{\Pi}{K} = \frac{L_I \left( \frac{a}{1-a} + \frac{c^*}{(1-a)(1-c^*)} \right) + k_c L_{II}}{t \frac{L_I}{(1-a)(1-c^*)}} = \frac{L_I \left( \frac{a}{1-a} + \frac{c^*}{(1-a)(1-c^*)} + \frac{k_c}{k} \right)}{t L_I \left[ \frac{1}{1-c^*} \cdot \frac{1}{1-a} \right]}$$

$$\text{We know also that: } k_c = \frac{k+c}{1-c} = \frac{k}{1-c} + \frac{c}{1-c} = k \frac{1}{1-c} + \gamma = k(1+\gamma) + \gamma$$

Thus if we replace $k_c$ by the above value $k(1+\gamma)+\gamma$, we get:

$$r_c(\text{prices}) = \frac{\alpha + (1+\alpha)\gamma^* + \frac{k(1+\gamma)+\gamma}{k}}{t[(1+\alpha)(1+\gamma^*)]} = \frac{k(\alpha + \gamma^* + \alpha\gamma^*) + k(1+\gamma)+\gamma}{kt(1+\alpha+\gamma^*+\alpha\gamma^*)}$$ $$r_c(\text{prices}) = \frac{k(1+\alpha+\gamma+\gamma^*+\alpha\gamma^*)+\gamma}{kt(1+\alpha+\gamma^*+\alpha\gamma^*)} \quad (15)$$

It is obviously different from the equation given in table 2 in Appendix 3, since the definition in this table corresponds to the Marxist definition, which is no longer the case here. In order to simplify this rather complicated expression, let us temporarily pose:

$$1 + \alpha + \gamma + \gamma^* + \alpha\gamma^* = a \quad (16)$$

$$1 + \alpha + \gamma^* + \alpha\gamma^* = b \quad (17)$$

We can now see that function $r_c = f(k)$ is still, as one might expect, a homographic function, because it is of the form:

$$r_c(\text{prices}) = f(k) = \frac{1}{t} \cdot \frac{ak + \gamma}{bk} \quad (18)$$
$$ \text{Its derivative is therefore } f'(k) = \frac{-b\gamma}{(tbk)^2} \qquad (19) $$

Since $b > 0$ and $t > 0$, it follows that the sign of this derivative is always negative, and that this function is therefore always decreasing, which implies that when we define the rate of profit in terms of prices, an increase in the primary rate of surplus-value is always reflected in a fall in the rate of profit, like it was the case for the rate of profit defined in terms of values.

To go further in studying function $r_c(\text{prices})$ would imply to go back to equation (15), which is quite complicated, and would therefore involve complex mathematical calculations. This explains that, rather than embarking on cumbersome reasoning, it seemed preferable to use a graphical representation of this function, giving ourselves as examples the value of some of the variables in its price expression involved in the evolution of $r_c$.

# 3. A graphical representation of the evolution of the rate of profit

## 3.1. The setting of parameters

To graphically represent function $r$ and function $r_c$, we need first to give values to the various parameters involved in the expression of these functions.

For the average lifespan of fixed capital $t$, we assume that it is 10 years, i.e. $t = 10$.

In addition, we consider that one-third of total fixed capital is used in section I, i.e. $a = \frac{1}{3}$ (going back to our usual definition of parameter $a$).

$$ \text{It follows that } \alpha = \frac{a}{1-a} = \frac{1/3}{2/3} = \frac{1}{2} = 0.5 $$

If we want to construct function $r_c(\text{values}) = f(k)$ for different values of $c$, already used in figures 20.2, we will also have different values for $\gamma = \frac{c}{1-c}$. At this point it must be understood that to be sure capitalists of section II can decide the price of the consumption goods that they sell and thus the amount of surplus-value and the overall consumption of capitalists, by setting parameter $c$ as a share of global consumption. But they cannot decide the value of $c_{II}$, i.e. their own share of this global consumption, because it depends on the price at which they buy fixed capital from capitalists of section I.

On the other hand, capitalists of section I can decide the price of fixed capital that they sell to those of section II by setting parameter $c^*$, which has been defined as a gross margin rate, i.e. the ratio of the gross margin allocated to consumption over the selling price of fixed capital. Through the setting of $c^*$ they can influence the value of $c_I$, but they cannot directly decide its precise value, i.e. their own share in global consumption, because it depends on $c$ and the
price of consumption goods. Similarly, and this clearly derives from equation (28) in chapter 12, $c_I$ also depends on $k$, the primary rate of surplus-value, and varies with it.

In fact, once the value of $c$, i.e. the share of all capitalists in global consumption, and thus the price of consumption goods, are set by capitalists in section II (which at the same time also set the rate of surplus-value $k_c$), capitalists of section I can obtain indirectly the share $c_I$ that they get in total consumption, by setting $c^*$ and thus the price of fixed capital sold to section II. Therefore the share $c_{II}$ obtained by capitalist of section II is logically determined afterwards, as the difference between $c$ and $c_I$, and must be considered as a residue. This being noted, and in order not to unduly complicate this example, let us make the assumption that $c^*$ is equal to 0.25 and does not change when $c$ changes.

We thus get for parameters $\gamma$ et $\gamma^*$ the following values:

For $c = 0.10$, we have $\gamma = 0.1111$

For $c = 0.15$, “ “ $\gamma = 0.1765$

For $c = 0.20$, “ “ $\gamma = 0.25$

For $c = 0.25$, “ “ $\gamma = 0.3333$

For $c = 0.30$ “ “ $\gamma = 0.4286$

For $c^* = 0.25$, we have $\gamma^* = 0.333$

For each of the five cases corresponding to the above parameters, the corresponding functions are defined, first in terms of values and secondly in terms of prices.

## 3.2. A graphical representation of function $r_c$ (values).

This function has the very simple general form $r_c = \frac{k+c}{kt}$, since it is not influenced by the distribution of surplus-value in the form of monetary profits among capitalists of both sections, which is mediated by rate $c^*$. It only depends on parameters $c$ and $t$, and obviously on $k$, the primary rate of surplus-value. Therefore it can be written for the five cases above:

Case 1: $r_c (values) = \frac{k+0.10}{10k}$ for $c = 0.10$

Case 2: $r_c (values) = \frac{k+0.15}{10k}$ for $c = 0.15$

Case 3: $r_c (values) = \frac{k+0.20}{10k}$ for $c = 0.20$

Case 4: $r_c (values) = \frac{k+0.25}{10k}$ for $c = 0.25$

Case 5: $r_c (values) = \frac{k+0.30}{10k}$ for $c = 0.30$
These five functions, with $r_c$ defined in terms of values, correspond to the five values of parameter $c$ shown above, and are represented in figure 21.1, as well as a horizontal line, $r_c = 0.10$, which represents the constant rate of profit (equal to $\frac{1}{t}$, with $t=10$) in the absence of capitalist consumption ($c = 0$).

This line is also the asymptote towards which the different curves tend, when the rate of primary surplus-value $k = L_I / L_{II}$ tends towards infinity.

Three vertical lines have also been drawn, which correspond to rates of primary surplus-value of 20%, 25% and 30% respectively. Therefore the figure is as follows:

<figure><img src="image_10.png" /><figcaption>Figure 21.1 - Evolution of the rate of profit $r_c$ with its current definition and in terms of values as a function of the primary rate of surplus-value $k$, for different values of $c$.</figcaption></figure>

We can easily observe from figure 21.1 that the rate of profit decreases with the increase in the rate of surplus-value, for any given value of parameter $c$. However, this last parameter exerts a strong influence on the rate of profit: the higher is $c$ and the higher is the profit rate. For instance, with a rate of primary surplus-value $k$ equal to 20%, we have a profit rate of 15% with $c$ equal to 10%, and of 25%, with $c$ equal to 30%. For $k$ equal to 30%, the profit rate goes from 13.33% for $c$ equal to 10%, to 20% for $c$ equal to 30%.

One could nevertheless wonder whether a rate of profit expressed in terms of value has any theoretical interest, since in the real world this rate is always calculated in terms of prices. In fact it is an interesting variable, because its amount depends only on the distribution of surplus-value between workers and capitalists as a whole, and whatever the rate of profit in terms of prices, once $k$ and $c$ are given, this rate will not change. On the contrary, the rate of profit defined in terms of prices also depends on $c^*$, i.e. on the price of fixed capital and therefore on the distribution of surplus-value and of consumption goods among capitalists.
### 3.3. A graphical representation for function $r_c$ (prices)

This function has the following general form:

$$r_c(\text{prices}) = f(k) = \frac{\Pi_c}{K} = \frac{k(1+\alpha+\gamma+\gamma^*+\alpha\gamma^*)+\gamma}{kt(1+\alpha+\gamma^*+\alpha\gamma^*)} \quad (\text{see equation 15 above})$$

For the five cases already defined, each one of them corresponding to a different value of parameter $c$, and with again $t=10$, $a = \frac{1}{3}$ (hence $\alpha = 0.5$), and $c^* = 0.25$ (hence $\gamma^* = 0.333$), function $r_c(\text{prices})$ becomes therefore, with all calculations made:

$$\text{Case 1: } r_c(\text{prices}) = \frac{2.111k + 0.111}{20k} \quad \text{for } c = 0.10$$

$$\text{Case 2: } r_c(\text{prices}) = \frac{2.176k + 0.176}{20k} \quad \text{for } c = 0.15$$

$$\text{Case 3: } r_c(\text{prices}) = \frac{2.25k + 0.25}{20k} \quad \text{for } c = 0.20$$

$$\text{Case 4: } r_c(\text{prices}) = \frac{2.333k + 0.333}{20k} \quad \text{for } c = 0.25$$

$$\text{Case 5: } r_c(\text{prices}) = \frac{2.429k + 0.429}{20k} \quad \text{for } c = 0.30$$

We must stress the fact that $\gamma$, and therefore $c$ (from which $\gamma$ is defined) do not appear in the denominator of the above expression defining $r_c(\text{prices}) = f(k)$, because at the denominator of equation (15) we only find $\gamma^*$, and therefore indirectly $c^*$ (from which $\gamma^*$ is defined). Combined with the assumption made above that a does not change and that $c^*$ is fixed and equal to 0.25, this explains that the denominator is always the same, and equal to $20k$.

The five curves corresponding to the five functions which were just defined appear in figure 21.2, and differ only from the differences in the magnitude of parameter $c$. They allow us to observe that an increase in the primary rate of surplus-value $k$ results in a decrease of the rate of profit: we have the same evolution in terms of prices as we had in terms of values. We can see also that this decline is slowing down once we exceed 0.20 for this rate $k$, and that all curves converge towards an asymptote which is represented by the straight line:

$$r_c = 0.1 = \frac{1}{t} \quad \text{when } k \text{ tends towards infinity.}$$
<figure><img src="image_11.png" /><figcaption>Figure 21.2 - Evolution of the rate of profit *r*<sub>c</sub> with its current definition and in terms of prices as a function of the primary rate of surplus-value *k*, for different values of *c*.</figcaption></figure>

One could say that this evolution to some extent confirms the validity of Marx's law, although within the framework of this graphical representation, we did no longer link the variation in the rate of profit to the organic composition of capital, but rather to the evolution of the rate of surplus-value, for the sake of simplicity. With this provision, the law interpreted in terms of surplus-value would be moreover better verified with this usual definition of the rate of profit, rather than with the Marxist definition including variable capital at the denominator of the fraction giving the rate of profit. On the basis of the usual definition of the rate of profit, its decrease is indeed verified for any value of c, and not only for the situations where $c > \frac{1}{1+t}$.

We can also observe that for a given rate of surplus-value, the rate of profit increases with the increase in $c$, which is denoted by the displacement of the curves upward and to the right, which accompanies this increase. Once more we can notice that the rate of profit is actually very sensitive to a rise in $c$: for instance with $k = 0.20$, the rate of profit almost doubles, going from 13.33% when $c$ is equal to 0.10, to 22.87% when $c$ is equal to 0.30. Capitalist consumption thus appears again to be of fundamental importance in explaining the evolution of the rate of profit.

Nevertheless, parameter $k$ maintains a strong influence, which for instance can be registered if we try to know what happens to $c_I$ and $c_{II}$, for a same value of $c$, for instance $c = 0.20$, when $k$ increases from 0.20 to 0.30. We know from equation (28) in chapter 12 that:
$$c_I = \gamma * \frac{k(1-c)}{(1+k)}$$

Therefore with $c = 0.20$, $c^* = 0.25$, and thus $\gamma^*=0.333$, we get:

1) For $k = 0.2$, which means that $k_c = \frac{k+c}{1-c} = 0.5 = 50\%$:

$$c_I = \gamma * \frac{k(1-c)}{(1+k)} = 0.333 \frac{0.2(1-0.2)}{(1+0.2)} = 0.333 \frac{0.16}{1.2} = 0.0444 = 4.44\% \text{ of total consumption}$$

Which implies that $c_{II} = c - c_I = 0.2 - 0.0444 = 0.1556 = 15.56\%$ of total consumption.

2) For $k = 0.30$, which means that $k_c = 0.625 = 62.5\%$, we get:

$$c_I = 0.333 \frac{0.3(1-0.2)}{(1+0.3)} = 0.333 \frac{0.24}{1.3} = 0.0615 = 6.15\%$$

Which implies that $c_{II} = c - c_I = 0.2 - 0.0615 = 0.1385 = 13.85\%$

This shows us that an increase in the primary rate of surplus-value, even with a constant share of consumption for all capitalists, results in an increase in the share of consumption for capitalists of section I, at the expense of the share of capitalists of section II.

This gives us a justification to examine more particularly function $k_c = f(c)$, which we will do in next subsection.

### 3.4. The relation between the actual rate of surplus value $k_c$ and the consumption of capitalists

The importance of capitalist consumption as a determinant of the rate of profit is due to the great elasticity of the actual rate of surplus-value $k_c$ with respect to $c$, which derives from the nature of the function defining $k_c$ in terms of $c$. This function is indeed nothing else than the definition of $k_c$:

$$k_c = f(c) = \frac{k+c}{1-c}, \text{ which as a function of } c \text{ should rather be rewritten now as:}$$

$$k_c = f(c) = \frac{c+k}{-c+1} \qquad (16)$$

$$\text{This is again a homographic function, whose derivative is } f'(c) = \frac{1+k}{(1-c)^2} \qquad (17)$$
This derivative is always positive, which implies that the rate of surplus-value $k_c$ is an increasing function of $c$, the share of capitalist consumption in total consumption.

Since $k_c = f(c)$ is a positive and differentiable function of a positive variable $c$, its elasticity at point $c$ is defined as:

$$Ef(c) = \frac{\frac{\delta k_c}{k_c}}{\frac{\delta c}{c}} = \frac{c}{f(c)} f'(c) = \frac{c(1-c)}{k+c} * \frac{1+k}{(1-c)^2} = \frac{c(1+k)}{(k+c)(1-c)} \quad (18)$$

We can calculate this elasticity for a given value of parameter $k$, taking for instance $k = 0.3$, and for two values of $c$, for instance for $c = 0.10$ and $c = 0.30$:

$$\text{For } c = 0.10, \text{ this elasticity is } Ef(c) = \frac{0.10(1+0.3)}{(0.3+0.1)0.9} = \frac{0.13}{0.36} = 0.36$$

$$\text{For } c = 0.30, \text{ the elasticity is } Ef(c) = \frac{0.30(1+0.3)}{(0.3+0.3)0.7} = \frac{0.39}{0.42} = 0.93$$

Through this simple example, we can see that the elasticity of $k_c = f(c)$ with respect to a relative change of $c$ increases very strongly when $c$ increases: this means that the higher is the share of capitalist consumption in total consumption, the stronger is the influence of its variation on the rate of growth of the rate of surplus-value $k_c$. This is reflected in the shape of the representative curves in the chart below, where they are plotted for four values of $k$, namely:

$k = 0.2, k = 0.3, k = 0.4 \text{ and } k = 0.5, \text{ respectively}$

This strong influence of $c$, which can be observed in the four curves drawn in figure 21.3, appears to be unsurprising, if we have a look at the second derivative of $k_c = f(c)$, which is:

$$f''(c) = \frac{2(1+k)}{(1-c)^3} \quad (19)$$
<figure><img src="image_12.png" /><figcaption>Figure 21.3 - Evolution of the rate of surplus-value $k_c$ with $k_c = f(c) = \frac{c+k}{-c+1}$</figcaption></figure>

Since we know that $c < 1$, by definition, we also have always $0 < (1-c)^3 < 1$. Since we have also $k > 0$, it follows that we always have $f''(c) > 2$.

The fact that $f''(c)=0$ is greater than 2 essentially implies that $f''(c)$ is always positive. In turn this means that $f'(c) = \frac{dk_c}{dc}$ is an increasing function of $c$. It follows that when $c$ increases, the variations of $k_c$, i.e. $\delta k_c$, are larger than the variations of $c$, i.e. $\delta c$. Relative variations of $k_c$, i.e. $\frac{\delta k_c}{k_c}$, therefore become larger than relative variations of $c$, i.e. $\frac{\delta c}{c}$. This means that the elasticity of $k_c$ with respect to $c$ rises when $c$ increases. This appears in the above example, where the elasticity is indeed more than twice as strong (going from 0.36 to 0.93) when $c$ goes from 0.10 to 0.30. This corresponds to the form taken by the representative curves of function $k_c = f(c)$ represented in figure 9 above, together with five vertical lines corresponding to different values of $c$ ($c = 0.10, c = 0.15, c = 0.20, c = 0.25, c = 0.30$).

Keeping again $k = 0.3$, we have for these different values of $c$:

$$1) k_c = f(c) = 0.44 \text{ for } c = 0.10$$

$$2) k_c = f(c) = 0.53 \text{ for } c = 0.15 \quad (\text{from 1) to 2) } \delta k_c = 0.090 \text{ and } \delta c = 0.05$$
$$3) k_c = f(c) = 0.625 \text{ for } c = 0.20 \quad \text{(from 2) to 3) } \delta k_c = 0,095 \text{ and } \delta c = 0.05$$

$$4) k_c = f(c) = 0.73 \text{ for } c = 0.25 \quad \text{(from 3) to 4) } \delta k_c = 0,105 \text{ and } \delta c = 0.05$$

$$5) k_c = f(c) = 0.86 \text{ for } c = 0.30 \quad \text{(from 4) to 5) } \delta k_c = 0,130 \text{ and } \delta c = 0.05$$

This evolution shows the rise in the values of the elasticity of $k_c$ compared to $c$, which becomes greater than one: indeed it reaches 1 for $0.324 < c < 0.325$.

### 3.5. A comparison between two expressions of the profit rate: in terms of values and in terms of prices

A final point worth mentioning is the great similarity of the two curves representing function $r_c$ as a function of $k$, one with the rate of profit defined on the basis of values, and the other where it is defined on the basis of prices. This similarity is apparent although their equations are significantly different, since for two values of parameter $c$, for example $c = 0.10$ and $c = 0.25$, these equations are as follows:

1) For $c = 0.10$, and keeping $\alpha = 0.5$, $t = 10$, and $\gamma* = 0.333$:

$$r_c(\text{values}) = \frac{k+c}{kt} = \frac{k+0,10}{10k};$$

$$r_c(\text{prices}) = \frac{k(1+\alpha+\gamma+\gamma^*+\alpha\gamma^*)+\gamma}{kt(1+\alpha+\gamma^*+\alpha\gamma^*)} = \frac{2.111k+0.111}{20k}$$

2) For $c = 0.25$:

$$r_c(\text{values}) = \frac{k+c}{kt} = \frac{k+0,25}{10k};$$

$$r_c(\text{prices}) = \frac{k(1+\alpha+\gamma+\gamma^*+\alpha\gamma^*)+\gamma}{kt(1+\alpha+\gamma^*+\alpha\gamma^*)} = \frac{2.333k+0.333}{20k}$$

This similarity appears in Figure 21.4, where the curves of the two functions, in terms of values and in terms of price, have been plotted for each of the two above values of $c$: $c = 0.10$ and $c = 0.25$. In both cases the proximity of the two curves is also striking.
<figure><img src="image_13.png" /><figcaption>Figure 21.4 - Comparison of curves representing $r_c$ in terms of values and prices</figcaption></figure>

This proximity is not very surprising, since profit is only surplus-value transformed and redistributed through exchanges that take place on the basis of monetary prices. Profit, therefore, is nothing else than the concrete form of surplus-value when it is monetized on the occasion of the realization of the product, which simultaneously ensures the realization of this surplus-value. We can also observe that the curves intersect. As a result, the price curves pass above the value curves when $k$ grows above amounts which may however be too high to correspond to the real world. However, this difference between the curves seems to increase slightly as $c$ increases because the curves then move upwards and to the right, which is related to the correlation between the price level of consumer goods and this rising capitalist consumption.

## 4. The evolution of the share of profits in the product

Paradoxically, all the analysis that has just been conducted on the rate of profit and its evolution, and in particular its dependence on and sensitivity to the evolution of many variables (the rate of surplus-value, certainly, but also the lifespan of fixed capital, as well as the consumption of different types of capitalists) tends to downplay the importance it can have, in the real world of capitalist enterprises, as an effective criterion of decision-making.

This is all the more true as the various rates of profit as they are calculated in practice by capitalist enterprises are highly dependent on tax and accounting rules, particularly those concerning the depreciation of fixed capital. The rate of profit in the real world is therefore a poor performance indicator, and this explains why financial analysts are not mistaken as regards its true significance. As mentioned previously these experts are actually much more
interested in the cash flow of a firm, which is referred to in financial jargon as EBITDA ("Earnings before interest, taxes, depreciation and amortization"). This concept is often used, moreover, to evaluate the value of a company in view of an acquisition. It corresponds in national accounts to the gross operating income, but also, in the form of a ratio, to what is called the margin rate.

The concept corresponds in the analysis conducted here to the ratio of total profits over total product, but as we know that this ratio has a fairly complex expression (see equation (52) in chapter 12 and table 2 in appendix 3), and especially that the realization of profits takes place according to a very different logic in each of the two sections, it seemed better to start by calculating the ratio for each of the two sections taken separately, before dealing with the global profit share. Let us call $\pi_I$ and $\pi_{II}$, these two ratios, respectively for section I and section II. We will show that each of these two ratios $\pi_I$ and $\pi_{II}$ is a function of different parameters: the evolution of $\pi_I$ depends on that of $a$ and $c^*$, while that of $\pi_{II}$ depends as already noted above on the evolution of $k$ and $c$.

### 4.1. The share of profits in the product of section I

In section I, to begin with, the profit / product ratio is equal to:

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

If we look at $\pi_I$ as a function of $a$, then we must write $\pi_I$ as above:

$$ \pi_I = f(a) = a(1-c^*)+c^* \qquad (20) $$

But $\pi_I$ can also be considered as a function of $c^*$ and in this case the function is written:

$$ \pi_I = f(c^*) = c^*(1-a)+a \qquad (21) $$

We can verify that both functions are linear functions, perfectly symmetrical to each other.

It seemed useful to provide a graphical representation of the corresponding functions, which is often more eloquent than long explanations.

1) Let us start with function $\pi_I = f(a) = a(1-c^*)+c^*$

It can be represented in a figure for different values of $c^*$, i.e. $c^* = 0.25$, $c^* = 0.30$, $c^* = 0.35$, $c^* = 0.40$ and $c^* = 0.45$. We recall that $c^*$ is a margin rate which corresponds to capitalist consumption in section I. As for the share of consumption of capitalists of section I in global consumption, it is $c_I$. The corresponding curves appear in figure 21.5, where we also add four straight lines corresponding to $a = 0.25$, $a = 0.30$, $a = 0.35$ and $a = 0.40$.
<figure><img src="image_14.png" /><figcaption>Figure 21.5 - Evolution of $\pi_I$ - the share of profits in section I - as a function of $a$ and for different values of parameter</figcaption></figure>

We can calculate that the share of profits is for instance 43.75 % for $c^* = 0.25$ and $a = 0.25$, and goes up to 67 % for $c^* = 0.45$ and $a = 0.40$.

Using the formula previously established in chapter 12 (page 235), which links $c^*$ and $c_I$,
i.e. $c_I = \gamma^* \frac{k(1-c)}{1+k}$, we can therefore calculate $c_I$, by taking for instance $k = 0.33$ and $c = 0.3$. Then we obtain:

$c_I = 5.83\%$, for $c^* = 0.25$

$c_I = 14.30\%$, for $c^* = 0.45$

2) Let us continue by representing now function $\pi_I = f(c^*) = c^*(1-a)+a$

It can also be represented for different values of $a$, i.e. $a = 0.20$, $a = 0.25$, $a = 0.30$, $a = 0.35$ and $a = 0.40$. We recall that $a$ is the share of the total value of produced fixed capital which is used in section I. The corresponding curves appear in figure 21.6, where we also add four straight lines corresponding to $c^* = 0.25$, $c^* = 0.30$, $c^* = 0.35$ and $c^* = 0.40$.

We can see that the share of profits is for instance 40% for $a = 0.20$ and $c^* = 0.25$ and goes up to 64 % for $a = 0.40$ and $c^* = 0.40$. We already know that for these last two values of $c^*$ (i.e. 0.25 and 0.40) we have $c_I = 5.83\%$ and $c_I = 14.30\%$.
<figure><img src="image_15.png" /><figcaption>Figure 21.6 - Evolution of $\pi_I$ - the share of profits in section I - as a function of $c^*$ and for different values of parameter $a$</figcaption></figure>

These two figures show us that the share of profits in section I is always increasing as a function of $a$ as well as a function of $c^*$ because the derivative of both functions is always positive, since by definition we always have $a < 1$ and $c^* < 1$. Moreover, both functions being of the same form, with a range of values quite similar, $\pi_I$ is quite similarly affected by changes of $c^*$ as well as by those of $a$. This shows once again the importance of capitalist consumption, here in section I, in the evolution of the share of profits, or reciprocally the fact that the share of profits plays a prominent role in the determination of capitalist consumption.

## 4.2. The share of profits in the product of Section II

In section II the profit / product ratio is equal to:

$$ \pi_{II} = \frac{\frac{L_{II}(k+c)}{1-c}}{\frac{L_{II}(1+k)}{1+c}} = \frac{k+c}{1+k} \quad (22) $$

The share of profits in this section is thus a function of only two variables: $k$, the primary rate of surplus-value, and $c$, the share of overall capitalist consumption in global consumption. In order to isolate the influence of each of these two variables, we can study two functions, the first one with $\pi_{II}$ as a function of $k$ and $c$ as a parameter having given values, and the second one with $\pi_{II}$ as a function of $c$ and $k$ as a parameter having given values.
1) Let us start first with $\pi_{II}$ as a function of $k$, which as such is:

$$ \pi_{II} = f(k) = \frac{k+c}{1+k} \qquad (23) $$

This is a homographic function. Its derivative is $f'(k) = \frac{1-c}{(1+k)^2}$, which is always positive, so that it is an increasing function of $k$, the primary rate of surplus-value.

We can provide a graphical representation of function $\pi_{II} = f(k)$ in figure 21.7. We do that by drawing five curves, each one corresponding to a different value of $c$:

$c = 0.10, c = 0.15, c = 0.20, c = 0.25 \text{ and } c = 0.30.$

We represent as well the straight vertical lines corresponding to $k = 0.25, k = 0.30, k = 0.35, k = 0.40$ and $k = 0.45$.

We can verify that the share of profits $\pi_{II}$ is for instance 28 % for parameter $c = 0.10$ and variable $k = 0.25$, and goes up to 51.7 % for $c = 0.30$ and $k = 0.45$.

<figure><img src="image_16.png" /><figcaption>Figure 21.7 - Evolution of $\pi_{II}$ - the share of profits in section II - as a function of $k$ and for different values of parameter $c$</figcaption></figure>

2) Let us continue now with $\pi_{II}$ as a function of $c$, which as such is:

$$ \pi_{II} = f(c) = \frac{k+c}{1+k} = \frac{1}{1+k}c + \frac{k}{1+k} \qquad (24) $$
It is a linear function whose derivative is $f'(c) = \frac{1}{1+k}$, a positive and constant number for a given value of $k$, so that it is an increasing function with a positive slope equal to $\frac{1}{1+k}$.

We can provide a graphical representation of function $\pi_{II} = f(c)$, in figure 21.8. We do that by drawing five curves, each one corresponding to a different value of $k$:

$k = 0.25, k = 0.30, k = 0.35, k = 0.40 \text{ and } k = 0.45.$

We represent as well the straight vertical lines corresponding to:

$c = 0.10, c = 0.15, c = 0.20, c = 0.25 \text{ and } c = 0.30.$

We can verify that the share of profits $\pi_{II}$ is obviously exactly the same for these different functions and for the same values of parameter $k$ and variable $c$: $\pi_{II} = 28\%$ for $k = 0.25$ and $c = 0.10$ and goes up to $\pi_{II} = 51.7\%$ for $k = 0.45$ and $c = 0.30$.

<figure><img src="image_17.png" /><figcaption>Figure 21.8 - Evolution of $\pi_{II}$ - the share of profits in section II - as a function of $c$ and for different values of parameter $k$</figcaption></figure>

All in all it is notable, if we compare these two last figures, that $\pi_{II}$, the share of profits in section II producing consumption goods, seems more affected by changes in $c$ than by those in $k$, which shows once again the importance of capitalist consumption as a whole as a key element in the functioning of the system.
### 4.3. The share of profits in the overall product

The expression giving the share of profits $\pi$ in the overall product is complicated, because of the two distinct modes of realization of profits that have been highlighted previously. It can nevertheless be calculated.

$$ 
\begin{aligned}
\pi &= \frac{\Pi}{Y} = \frac{L_I \left( \frac{a}{1-a} + \frac{c^*}{(1-a)(1-c^*)} \right) + L_{II} \frac{k+c}{1-c}}{\frac{L_I}{(1-a)(1-c^*)} + L_{II} \frac{1+k}{1-c}} = \frac{(\alpha + \gamma^* + \alpha\gamma^*) + \frac{1}{k} \left( \frac{k}{1-c} + \frac{c}{1-c} \right)}{1 + \alpha + \gamma^* + \alpha\gamma^* + \frac{1}{k} \left( \frac{1}{1-c} + \frac{k}{1-c} \right)} \\
\pi &= \frac{\Pi}{Y} = \frac{k(\alpha + \gamma^* + \alpha\gamma^*) + k(1+\gamma) + \gamma}{k(1 + \alpha + \gamma^* + \alpha\gamma^*) + 1 + \gamma + k(1+\gamma)} = \frac{k(1 + \alpha + \gamma + \gamma^* + \alpha\gamma^*) + \gamma}{k(2 + \alpha + \gamma + \gamma^* + \alpha\gamma^*) + 1 + \gamma} \quad (25)
\end{aligned}
$$

By putting $\Psi = 1 + \alpha + \gamma + \gamma^* + \alpha\gamma^*$, the expression becomes quite simplified, and we get:

$$ \pi = \frac{\Pi}{Y} = \frac{k\Psi + \gamma}{k(1+\Psi) + 1+\gamma} \quad (26) $$

$$ \text{If we now consider that } \pi = f(k) \text{ then its derivative is } f'(k) = \frac{\Psi(1+\gamma) - \gamma(1+\Psi)}{[k(1+\Psi) + 1+\gamma]^2} \quad (27) $$

$$ \text{The condition of positivity of the derivative is } \Psi(1+\gamma) - \gamma(1+\Psi) > 0 \quad (28) $$

When performing the calculation, it is immediate that this implies $\Psi - \gamma > 0$, which is always verified, given the definition of $\Psi$, whose value is always positive and includes $\gamma$. So the share of profits $\pi$ in the overall product $Y$ is always an increasing function of $k$, the primary rate of surplus-value.

The phenomenon is more meaningful by providing a graphical representation of function $\pi = f(k)$, which implies to give a value to its different parameters.

$$ \text{Let us first take } a = 0.25, \text{ which gives us } \alpha = \frac{a}{1-a} = \frac{0.25}{1-0.25} = 0.333. $$

We can then represent our function for different values of $c$, which we arbitrarily take as:

$$ c = 0.10, c = 0.15, c = 0.20, c = 0.25, \text{ and } c = 0.30, $$

These values of $c$ correspond to the following corresponding values for $\gamma = \frac{c}{1-c}$:

$$ \gamma = 0.111, \gamma = 0.177, \gamma = 0.250, \gamma = 0.333, \text{ and } \gamma = 0.429 $$

We can also assume that there are different values of $c^*$:
$c^*=0.25, c^*=0.30, c^*=0.35, c^*=0.40 \text{ and } c^*=0.45.$

We recall again that $c^*$ is a margin rate which corresponds to capitalist consumption in section I, and is linked to $c_I$, i.e. precisely the share of capitalists of section I in total consumption, by a formula previously established in chapter 12 (page 235), which is

$c_I = \gamma^* \frac{k(1-c)}{1+k}$. Since $c_I$ depends not only on $c^*$ (and thus $\gamma^*$), but also on $k$, its value will change with the level of $\pi = f(k)$ and cannot be considered as a parameter with a given value.

The values of $c^*$ correspond to the following corresponding values for $\gamma^* = \frac{c^*}{1-c^*}$:

$\gamma^*=0.333, \gamma^*=0.429, \gamma^*=0.538, \gamma^*=0.667, \text{ and } \gamma^*=0.811$

If we want now to give an example of a few numerical values for $\pi$, since $\pi$ is a function of $k$ we need to adopt a numerical value for $k$. Let us decide arbitrarily that we have $k = 20$. Then the corresponding values for all these parameters are given in table 21.1.

<table><caption>Table 21.1. Values of $\pi$ corresponding to different values of the other parameters</caption><thead><tr><th>α</th><th>c</th><th>γ</th><th>c*</th><th>γ*</th><th>αγ*</th><th>Ψ</th><th>k</th><th>π</th></tr></thead><tbody><tr><td>0.333</td><td>0.10</td><td>0.111</td><td>0.25</td><td>0.333</td><td>0.111</td><td>1.888</td><td>0.20</td><td><strong>0.290</strong></td></tr><tr><td>0.333</td><td>0.15</td><td>0.177</td><td>0.30</td><td>0.429</td><td>0.143</td><td>2.082</td><td>0.20</td><td><strong>0.331</strong></td></tr><tr><td>0.333</td><td>0.20</td><td>0.250</td><td>0.35</td><td>0.538</td><td>0.179</td><td>2.300</td><td>0.20</td><td><strong>0.372</strong></td></tr><tr><td>0.333</td><td>0.25</td><td>0.333</td><td>0.40</td><td>0.667</td><td>0.222</td><td>2.555</td><td>0.20</td><td><strong>0.413</strong></td></tr><tr><td>0.333</td><td>0.30</td><td>0.429</td><td>0.45</td><td>0.811</td><td>0.270</td><td>2.843</td><td>0.20</td><td><strong>0.454</strong></td></tr></tbody></table>

We get for $\pi$ a range of values going from 29 % to 45.4 %, which are quite plausible.

If we now want to see what happens to $\pi$ when $k$ is no longer set at $k = 0.20$, but becomes a variable, an appropriate way is to give a graphical representation of $\pi = f(k)$. If we keep constant all the other variables, i.e. $\alpha, c, \gamma, c^*, \gamma^*$ and $\Psi$ which appear on each line of the above table we are able to define five distinct functions $\pi = f(k)$.
These five functions are the following ones:

$$1) \pi = (1,888 k + 0,111)/(2,888 k + 1,111)$$

$$2) \pi = (2,082 k + 0,177)/(3,082 k + 1,177)$$

$$3) \pi = (2,30 k + 0,250)/(3,3 k + 1,25)$$

$$4) \pi = (2,555 k + 0,333)/(3,555 k + 1,333)$$

$$5) \pi = (2,843 k + 0,429)/(3,843 k + 1,429)$$

We can now represent these five functions under the form of five curves, each of them corresponding to a line in table 1 above, save obviously for the values of $k$ and $\pi$. They appear in figure 21.9.

<figure><img src="image_18.png" /><figcaption>Figure 21.9 – Evolution of the share of profits in the product $\pi$ as a function of $k$</figcaption></figure>

The five curves shown above are representing function $\pi = f(k)$ for five different values of parameters $c$ and $c^*$. This figure shows clearly that the share of profits in the product, i.e. the overall gross margin rate in a macroeconomic sense, is an increasing function of the primary rate of surplus-value, and therefore varies inversely with the rate of profit, which is itself a decreasing function of this primary rate of surplus-value. Indeed chart 8 in subsection 3.3 of this chapter showed that in a graphical way. This last result might seem therefore somewhat counterintuitive, if we did not remember that an increase in $k$ causes an increase in the organic composition of capital which, as we also showed at length in the preceding sections of this same chapter, always results in a reduction of the rate of profit.
Another reason for this dissimilar evolution comes from the fact that the rise in the share of profits results ipso facto from a reduction in the share of wages which is at the origin of a rise both in the rate of surplus-value $k_c$, through an increase in $c$, the share of capitalist consumption in total consumption, and in the organic composition of capital $q_c$, itself inducing also, as we just recalled, a reduction in the rate of profit.

To close this chapter, we may now briefly devote a last sub-section to show that the share of profits is obviously linked to the rate of profit, the latter being - as we can easily imagine, a growing function of the former.

### 4.4. The share of profits and the rate of profit

The relationship between these two variables it quite easy to establish, since we know that by definition:

$$r_c = \frac{\Pi}{K} = \frac{\Pi}{Y} \cdot \frac{Y}{K} = \pi \frac{Y}{K} \qquad (29)$$

We will not repeat the calculation of $Y$ and $\Pi$, since it has already been carried out in previous sections, and therefore will go directly to its result, which gives us:

$$r_c = \pi \left[ \frac{k(2+\alpha+\gamma+\gamma^*+\alpha\gamma^*)+1+\gamma}{kt(1+\alpha+\gamma^*+\alpha\gamma^*)} \right] \qquad (30)$$

In order to simplify this expression, we take again $\Psi = 1+\alpha+\gamma+\gamma^*+\alpha\gamma^*$

This ultimately gives us the following expression for $r_c = f(\pi)$:

$$r_c = \pi \left[ \frac{k(1+\Psi)+1+\gamma}{kt(\Psi-\gamma)} \right] \qquad (31)$$

It is immediate to observe that the rate of profit $r_c = f(\pi)$ is an increasing function of $\pi$. But we can observe one more time that it remains also a growing function of $a$, $c$ and $c^*$, and a decreasing function of $k$, the primary rate of surplus-value. To give a few examples showing the evolution of the rate of profit in parallel with the share of profits, we can use the values of the various parameters appearing in table 21.1 of previous sub-section 4.3., with all of them corresponding again to a unique value of $k$, given as $k = 0.20$.

We can see that the increase in the share of profits which is displayed in this table is necessarily due to the increase in $k_c$ (since $k$ is a constant, as well as $a$). It has thus been deemed useful to also indicate the effective rate of surplus-value $k_c = \frac{k+c}{1-c}$. This allows us to provide table 21.2, which gives all the values of the parameters for each line, and thus for each value of $\pi$ that had been calculated.
<table><caption>Table 21.2 Values of $r_c = f(\pi)$ corresponding to different values of the parameters</caption><thead><tr><th>α</th><th>c</th><th>γ</th><th>c*</th><th>γ*</th><th>αγ*</th><th>Ψ</th><th>k</th><th>k<sub>c</sub></th><th>π</th><th>r<sub>c</sub></th></tr></thead><tbody><tr><td>0.333</td><td>0.10</td><td>0.111</td><td>0.25</td><td>0.333</td><td>0.111</td><td>1.888</td><td>0.20</td><td>0.333</td><td><strong>0.290</strong></td><td><strong>0.138</strong></td></tr><tr><td>0.333</td><td>0.15</td><td>0.177</td><td>0.30</td><td>0.429</td><td>0.143</td><td>2.082</td><td>0.20</td><td>0.412</td><td><strong>0.331</strong></td><td><strong>0.156</strong></td></tr><tr><td>0.333</td><td>0.20</td><td>0.250</td><td>0.35</td><td>0.538</td><td>0.179</td><td>2.300</td><td>0.20</td><td>0.50</td><td><strong>0.372</strong></td><td><strong>0.181</strong></td></tr><tr><td>0.333</td><td>0.25</td><td>0.333</td><td>0.40</td><td>0.667</td><td>0.222</td><td>2.555</td><td>0.20</td><td>0.60</td><td><strong>0.413</strong></td><td><strong>0.190</strong></td></tr><tr><td>0.333</td><td>0.30</td><td>0.429</td><td>0.45</td><td>0.811</td><td>0.270</td><td>2.843</td><td>0.20</td><td>0.714</td><td><strong>0.454</strong></td><td><strong>0.207</strong></td></tr></tbody></table>

For $\pi = 0.290$, we have $r_c = f(\pi) = 0.138 = 13.8\%$

For $\pi = 0.331$, we have $r_c = f(\pi) = 0.156 = 15.6\%$

For $\pi = 0.372$, we have $r_c = f(\pi) = 0.181 = 18.1\%$

For $\pi = 0.413$, we have $r_c = f(\pi) = 0.190 = 19.0\%$

For $\pi = 0.454$, we have $r_c = f(\pi) = 0.207 = 20.7\%$

It is easy to see that $r_c = f(\pi)$ is an increasing function of $\pi$, for a given value of $k$, the primary rate of surplus-value, which here is $k = 0.20$. Moreover these values of $r_c$, ranging from 13.8% to 20.7%, are themselves quite plausible. Obviously these values of $r_c$ would be lower with higher values of $k$ and of $q_c$, the organic composition of capital. In any case, what these calculations and examples show clearly is once more that the main driving force behind the increase in the share of profits as well as in the rate of profit is the increase in the rate of surplus-value resulting from the increase in capitalist consumption, which depends on the level of prices in section II producing consumption goods.

The confirmation of the prominent role played by this parameter justifies again the importance given to a model taking into account capitalist consumption. The conclusion that we can draw from the thorough examination of these models, both with the Marxist and current definition of the rate of profit, tends finally to minimize the role played by this variable as such, because it has been shown first that it was before all a statistical average of multiple rates, and second that even its level as an average was influenced by a number of various parameters. Some of these parameters are of a technical and exogenous nature, like $k$, $t$ and $a$. But it is clearly another one, $c$, as the driving force of the rate of surplus-value $k_c$, which is the pivot of the functioning of the whole economic system. Its determination is at the
heart and the outcome of a social conflict for the distribution of consumption goods between
workers and capitalists, which goes far beyond the realm of economic theory alone.

This examination of the evolution of the rate of profit in relation to various variables - including the share of profits in the product, is now complete, we can now formulate two new principles.

**Principle 43:** When then the rate of profit defined according to its current definition, (i.e. without applying this rate to wages - as if they were advanced), and in a situation of simple reproduction with capitalist consumption, this rate is an increasing function of *k*, the primary rate of surplus-value, *k*<sub>c</sub>, the actual rate of surplus-value with capitalist consumption, and *c*, the share of capitalist consumption in total consumption. On the contrary it is decreasing in terms of *q*<sub>c</sub>, the organic composition of capital. This confirms the validity of the Marxist law of the tendency of the rate of profit to fall when the organic composition of capital increases. But this law remains a mere tendency, since it can be thwarted by an increase in capitalist consumption, which shows the paramount importance of this variable.

**Principle 44:** The share of profits in the product (named π) is also an increasing function of the primary rate of surplus-value and of the share of capitalist consumption, and thus necessarily of *k*<sub>c</sub>, the actual rate of surplus-value. The rate of profit itself is an increasing function of π.

However, it must be remembered that the models that we used are valid only for the analysis
of simple reproduction. We will therefore look in the following and last chapter of this book
at some of the elements that make it possible to analyze extended reproduction. The
complexity of the phenomenon, however, will lead us not to engage in the construction of a
full-fledged new model and rather to restrain ourselves to a numerical example and some
preliminary considerations.

**References**

Flamant, Christian (2015a). 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.

Piketty, Thomas (2013). *Le capital au XXI<sup>ème</sup> siècle*. Paris: Editions du Seuil. English edition: Piketty, T. (2014). *Capital in the Twenty-First Century* (translation: Goldhammer, A.). Cambridge (USA): Harvard University Press.
**Chapter 22. Some reflections and first elements of discussion to introduce extended reproduction**

Extended reproduction is typically a Marxist concept, which is dealt with in volume II (or book II) of Capital, published by Engels in 1885, after Marx’s death. In this chapter we will recall briefly in a first section how this concept is introduced by Marx, before trying to clarify some questions of definitions. Then, since developing a complete model of extended reproduction would be quite a complicated task, we will confine ourselves to the presentation of two numerical examples, the first one without capitalist consumption and the second one with capitalist consumption. Due to this Marxist point of departure, we will use in our examples the Marxist definition of the rate of profit, with variable capital – i.e. the value of labor-power, at the denominator of the expression defining this variable.

**1. Extended reproduction in Marx's Capital**

Marx touches upon the matter on three occasions in volume II of Capital. First in part II “Accumulation and reproduction on an extended scale” of chapter 2, where he indicates: “Accumulation, or production on an extended scale, which appears as a means for constantly more expanded production of surplus-value - hence for the enrichment of the capitalist, as his personal aim - and is comprised in the general tendency of capitalist production, becomes later, however, as was shown in Book I, by virtue of its development, a necessity for every individual capitalist” (Marx, 1885, p. 45).

It must be understood that accumulation here means an increase in the stock of constant capital, which for Marx includes both fixed capital and circulating capital.

Marx deals again with the same question in chapter 17 “The Circulation of Surplus-Value”, where the second part has the same title as above and deals essentially with the circulation of money associated with the accumulation of additional constant capital (on top of what is needed in the case of simple reproduction). This explains that he starts by writing: “Accumulation takes place in the form of extended reproduction, it is evident that it does not offer any new problem with regard to money-circulation” (Marx, 1885, p. 209).

Finally, it is in chapter 21, the very last of volume II, again under the same heading, that Marx really addresses the matter, through reproduction schemes, and in a way which is however quite complicated, because he continues to mix it with the problem of the origin of the additional money required to finance additional investments in constant capital, on top of what is needed in simple reproduction.

Contrarily to Marx, we do not think that there is any specific problem of realization in a situation of extended reproduction, mostly because we are not reasoning within the framework of his mainly wrong theory of money, which created a lot of problems in his
theory. Moreover, as we showed in chapter 11, Marx's reproduction schemes are erroneous, and there is a detailed demonstration on this question in Appendix II.

## 2. Some questions of definitions

The models for the realization of the product that we exposed so far were based on an assumption of simple reproduction. We will now see what happens to these models if we abandon the simple reproduction hypothesis, in order to integrate extended reproduction into them, as an increase in the rate of surplus-value leading to additional investments and an increase in the stock of fixed capital, which can be accompanied (or not) by a growth in capitalist consumption. It should be understood, however, that rejecting the assumption that fixed capital transmits its value to the product's must lead to redefine what Marx names simple reproduction and extended reproduction.

Indeed for Marx, in the case of simple reproduction, there could not be any surplus-value and therefore any accumulation of capital, as long as there would not be any consumption of capitalists, since in such a situation it is the value transmitted by the fixed capital that is already in use, and not an equivalent amount of surplus-value, which replaces the value lost by this same fixed capital. There would be no growth either, since Marx does not seem to imagine that with a fixed capital with unchanged characteristics there can be any increase in the (physical) productivity of labor, for instance thanks to a better use of existing fixed capital, or by a better organization of the production process with this unchanged fixed capital.

However we have already pointed out that, as soon as this hypothesis of transmission of the value of fixed capital is abandoned, there is necessarily a surplus-value, even without any consumption of capitalists, because it is this surplus-value, and not the value supposed to be "lost" by the fixed capital in operation, which – once monetized in the form of profits, is used to acquire the newly produced fixed capital due to replace the portion which leaves the production process at the end of its lifespan.

Similarly for Marx, extended reproduction corresponds to the accumulation of capital by accumulation of surplus-value, beyond the simple replacement of existing fixed capital, and only in this case can there be for him economic growth in the trivial sense of the term, that is to say growth of the quantities that are produced. Expanded reproduction is therefore in fact the growth of the stock of constant capital by accumulation of surplus-value, and this accumulation – excluding capitalists consumption by definition - has for counterpart only an investment which is net of constant capital replacement, and for fixed capital an investment which is net of the replacement of amortized capital.

In the present work, the rejection of the assumption of the transfer of value of fixed capital to the product implies that extended reproduction is defined by the accumulation of additional surplus-value, beyond the surplus-value already present in simple reproduction and corresponding to a mere replacement or maintenance of the value of fixed capital. This means that extended reproduction is therefore defined by an increase in the value of fixed capital
produced by section I, which by itself does not necessarily imply a parallel increase in the
overall value of consumer goods produced by section II. Indeed, if the increase in the stock of
fixed capital is shared between section I and section II, the latter may be able to increase the
physical quantities that it produces, without necessarily having to hire additional workers - i.e.
without a rise in the value produced in section II, particularly if the newly invested additional
fixed capital saves labor time by increasing its physical productivity. Such a phenomenon also
ensures that this accumulation of additional fixed capital is accompanied simultaneously by
an increase in the organic composition of capital. A similar result might nonetheless be
obtained with a faster rate of increase of the value produced in section I compared with that
produced in section II.

As already indicated, we will not try to provide a complicated model for this situation of
extended reproduction, and will rather present in the next sections two simple numerical
examples, which will be enough to shed some light on what is of interest for us, and which is
not the realization of the product, but the evolution of the main variables, like the rate of
surplus-value and capitalist consumption, and their influence on the profit rate.

3. A first numerical example, without capitalist consumption

In order to figure out what can be the evolution of the average rate of profit in a situation of
extended reproduction, we will use a numerical example already exposed above in section 3
of chapter 19 (see pp. 365-366) by looking now at the dynamic sequence going from a first
situation of simple reproduction (A), to a second one (B) characterized by an organic
composition of capital which is double that of the first one, and a rising rate of profit, which
goes from 9.09% to 11.1 %.

Let us assume, for example, that we gradually move from an annual fixed capital investment
with a value of 100, in this situation A of a simple reproduction regime allowing only for the
replacement of fixed capital at the end of its lifespan (supposed to be 7 years), to an annual
capital accumulation of 200, and this in five years, the newly accumulated capital during these
successive years having a value of 120, 140, 160, 180 and 200. This means that during the
first year the value of the means of production produced with previously accumulated capital
increases by 20 %, a priori because of the employment of 20 % of additional workers, all
other things being equal. During the second period the increase is 16.7 % and so on with
decreasing rates until the value of the newly produced means of production reaches the level
200 at the end of the fifth year.

Taking into account the convention adopted for the decommissioning of fixed capital, and
always assuming a seven-year lifespan for this capital, it is necessary to wait until an
additional period of seven years has elapsed before we reach a new stable situation, where the
totality of the accumulated capital corresponds entirely to the new level attained by yearly
investments, and to end up in a situation B of simple reproduction, which constitutes the new
cruise regime. The transition from the first simple reproduction regime to the second one,
marked by an organic composition of capital which has doubled, thus takes place in a twelve
year period. The evolution of the rate of profit and of the different variables which determine it appear in table 22.1.

<table><caption>Table 22.1 - A numerical example tracing the evolution of the rate of profit with extended reproduction</caption><thead><tr><th>Years</th><th>0</th><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th><th>6</th><th>7</th><th>8</th><th>9</th><th>10</th><th>11</th><th>12</th></tr></thead><tbody><tr><td>V = L<sub>II</sub><br>Value of labor force</td><td>400</td><td>400</td><td>400</td><td>400</td><td>400</td><td>400</td><td>400</td><td>400</td><td>400</td><td>400</td><td>400</td><td>400</td><td>400</td></tr><tr><td>L<sub>I</sub> = S<br>value of capital<br>accumulated during<br>each period</td><td>100</td><td>120</td><td>140</td><td>160</td><td>180</td><td>200</td><td>200</td><td>200</td><td>200</td><td>200</td><td>200</td><td>200</td><td>200</td></tr><tr><td>k = S/V<br>Rate<br>of surplus-value</td><td>25 %</td><td>30 %</td><td>35 %</td><td>40 %</td><td>45 %</td><td>50 %</td><td>50 %</td><td>50 %</td><td>50 %</td><td>50 %</td><td>50 %</td><td>50 %</td><td>50 %</td></tr><tr><td>K<br>Value of accumulated<br>Fixed capital</td><td>700</td><td>700</td><td>720</td><td>760</td><td>820</td><td>900</td><td>1000</td><td>1100</td><td>1200</td><td>1280</td><td>1340</td><td>1380</td><td>1400</td></tr><tr><td>q = K/V<br>Organic composition<br>of capital</td><td>175</td><td>175</td><td>1.8</td><td>1.9</td><td>2.05</td><td>2.25</td><td>2.5</td><td>2.75</td><td>3.0</td><td>3.2</td><td>3.35</td><td>3.45</td><td>3.5</td></tr><tr><td>K + V<br>capital advanced<br>(Marxist definition)</td><td>1100</td><td>1100</td><td>1120</td><td>1160</td><td>1220</td><td>1300</td><td>1400</td><td>1500</td><td>1600</td><td>1680</td><td>1740</td><td>1780</td><td>1400</td></tr><tr><td>S/V<br>Average rate of<br>profit</td><td>9.1 %</td><td>10.9 %</td><td>12.5 %</td><td>13.8 %</td><td>14.8 %</td><td>15.4 %</td><td>14.3 %</td><td>13.3 %</td><td>12.5 %</td><td>11.9 %</td><td>11.5 %</td><td>11.2 %</td><td>11.1 %</td></tr></tbody></table>
At first glance one might think that function $r = \frac{q}{(q+1) \cdot t}$ remains valid to describe the evolution of the rate of profit during this twelve year period, since it goes from 9.1% to 11.1%, which confirms that within the framework of the simplified model without capitalist consumption the Marxist law does not hold, because the rise in the organic composition of capital is accompanied by a rise in the rate of profit. But the numerical values provided by this example lead to note also that the rate of profit does not rise from 9.1 % to 11.1 % following a steady progression. On the contrary this rate goes through a maximum of 15.4 % in the fifth year, then goes down, first quickly and then more slowly, to arrive to the value of 11.1% corresponding to the new situation of simple reproduction.

This evolution of $r$ is thus no longer monotonous and strictly increasing and therefore cannot be explained by function $r = \frac{q}{(q+1) \cdot t}$, which as such does not account for the evolution of the rate of profit between the two simple reproduction regimes observed during year 0 and year 12. But this should not be surprising, if we remember that the definition of this function was developed on the basis of the explicit assumption of a very simple linear relation between the quantities of fixed capital $K$ and of surplus-value $L_I$, since it has been stated that $K = \sum_{i=0}^{i=t} K_i = L_I t$. Such a relationship therefore also exists between the rate of surplus value $k$ and the organic composition of capital, since $q = kt$. If $t$ is a constant, as is the case in this example, this relation becomes even a pure and simple identity ($q = 7k$). It can easily be checked for instance for years 0 and 12.

Now in the present numerical example, this relation $q = kt$ is no longer satisfied, and $t$ is no longer a simple scale factor that cannot influence the variation of function $r = f(q)$, since we do not pass instantly from situation A to situation B. In the meantime, and as it is easy to observe by examining the rate of surplus-value and organic composition in years 1 to 11, and even if $t$ remains constant ($t = 7$ years), this relation is not verified, and it is observed on the contrary that $q \neq kt$. This is so because the value of the accumulated fixed capital cannot move instantly from one level reflecting a stable regime of the rate of surplus-value to another stable regime, but does so over a period which is all the more extensive (12 years in this example), that the average lifespan of fixed capital is long. There is thus an inertia in the value of the stock of capital, which, because of the seven-year lifespan of each generation of accumulated capital, evolves much more slowly than surplus-value and the rate of surplus-value.

By expressing the evolution of the value of the capital stock $K$ according to the value of additional capital $I_i$ accumulated during the preceding periods (and equal in this example to the surplus-value of each period), it is of course theoretically possible to write a mathematical formula expressing this evolution within this framework: $K = f(I_i) = f(k)$.
This formula would take the form of a linear equation of recurrence, and one could introduce it into the function giving the rate of profit, which would become greatly complicated. But within the framework of this book, this type of mathematical refinement does not seem useful. This is the reason why we kept to a simple numerical example, which appears sufficient for the lessons that one wishes to obtain from it.

What other lessons can we draw from this example? First of all, that the process of extended accumulation allows the profit rate to exceed the limit previously highlighted of $r = \frac{1}{t}$, which for a fixed asset life of 7 years is $r = \frac{1}{t} = 14.29\%$. This limit cannot be crossed in a situation of simple reproduction, no matter how great the organic composition of capital. The transition from a simple reproduction situation to another one marked by a higher organic composition is thus accompanied by a fluctuation in the rate of profit during which it exceeds the level at which it will eventually stabilize, a level which is nevertheless higher than the initial one.

A second lesson is that the rate of profit, if not volatile, is at least more subject to fluctuations than the organic composition of capital. However, these fluctuations are much more moderate than the change in the rate of surplus-value, because of the buffer effect played by the stock of fixed capital. The numerical example shows us that a doubling of the rate of surplus-value results only in a limited increase in the rate of profit, which goes from 9.1% to only 11.1%, because of the concomitant doubling of the organic composition of capital.

## 4. A second example, with capitalist consumption

In this example, and as can be seen from Table 22.2, the economy is supposed to be in year 0 in a simple reproduction regime, with a capitalist consumption of 60, corresponding to a ratio $c$ of 60/260 or 23.1% of the value of consumer goods, which is 260. Since the amount of fixed capital accumulated in each period is supposed to have a value of 50, surplus-value is therefore 60 + 50, i.e. 110, and as the value of the labor force is $V_c = L_{II} - C_c = L_{II}(1-c) = 260(1-0.231) = 200$, the rate of surplus-value is at the outset 110/200 or 55%. Assuming now a lifespan of 10 years for fixed capital (3 years more than in the previous example), the amount of accumulated capital is therefore $50*10 = 500$. The profit rate is thus 15.7%.

Then the system enters in year 1 and up to year 14 in a process of gradual and simultaneous increase in the consumption of both workers (+10 each year) and capitalists (+5 each year), on the one hand, and of investment $I_i$ (+5 each year), on the other hand. Thus after 14 years, during which the amount of capitalist consumption has thus reached 130 and that of workers consumption 340 – corresponding to the value of the labor force $V_c$, the system stabilizes at this level. Capitalist consumption has thus risen from 23.1% to 130/470, or 27.7% of the value of consumer goods. Annual investment has increased from 50 to 120. Surplus-value has thus reached 250, and the rate of surplus-value 250/340, or 74%, instead of 55% initially.
<table><caption>Table 22.2 - A numerical example tracing the evolution of the rate profit with extended reproduction and capitalist consumption</caption><thead><tr><th>Years</th><th>0</th><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th><th>6</th><th>7</th><th>8</th><th>9</th><th>10</th><th>11</th><th>12</th><th>13</th><th>14</th><th>...</th><th>23</th></tr></thead><tbody><tr><td>Value of labor force L<sub>I</sub> = V<sub>c</sub></td><td>200</td><td>210</td><td>220</td><td>230</td><td>240</td><td>250</td><td>260</td><td>270</td><td>280</td><td>290</td><td>300</td><td>310</td><td>320</td><td>330</td><td>340</td><td>...</td><td>340</td></tr><tr><td>L<sub>I</sub> = Ii Value of capital accumulated in each period</td><td>50</td><td>55</td><td>60</td><td>65</td><td>70</td><td>75</td><td>80</td><td>85</td><td>90</td><td>95</td><td>100</td><td>105</td><td>110</td><td>115</td><td>120</td><td>...</td><td>120</td></tr><tr><td>Consumption of capitalists</td><td>60</td><td>65</td><td>70</td><td>75</td><td>80</td><td>85</td><td>90</td><td>95</td><td>100</td><td>105</td><td>110</td><td>115</td><td>120</td><td>125</td><td>130</td><td>...</td><td>130</td></tr><tr><td>S<sub>c</sub> Surplus-value</td><td>110</td><td>120</td><td>130</td><td>140</td><td>150</td><td>160</td><td>170</td><td>180</td><td>190</td><td>200</td><td>210</td><td>220</td><td>230</td><td>240</td><td>250</td><td>...</td><td>250</td></tr><tr><td>k<sub>c</sub>=S<sub>c</sub>/V<sub>c</sub> rate of surplus-value</td><td>55%</td><td>57%</td><td>59%</td><td>61%</td><td>63%</td><td>64%</td><td>65%</td><td>67%</td><td>68%</td><td>69%</td><td>70%</td><td>71%</td><td>72%</td><td>73%</td><td>74%</td><td>...</td><td>74%</td></tr><tr><td>K Value of total fixed capital</td><td>500</td><td>505</td><td>515</td><td>530</td><td>550</td><td>575</td><td>605</td><td>640</td><td>680</td><td>725</td><td>775</td><td>825</td><td>875</td><td>925</td><td>975</td><td>...</td><td>1200</td></tr><tr><td>q<sub>c</sub> = K/V<sub>c</sub> Organ. composition Of capital</td><td>2.50</td><td>2.40</td><td>2.34</td><td>2.30</td><td>2.29</td><td>2.30</td><td>2.33</td><td>2.37</td><td>2.43</td><td>2.50</td><td>2.58</td><td>2.66</td><td>2.73</td><td>2.80</td><td>2.87</td><td>...</td><td>3.53</td></tr><tr><td>K + V<sub>c</sub> Capital advanced</td><td>700</td><td>715</td><td>735</td><td>760</td><td>790</td><td>825</td><td>865</td><td>910</td><td>960</td><td>1015</td><td>1075</td><td>1135</td><td>1195</td><td>1255</td><td>1315</td><td>...</td><td>1540</td></tr><tr><td>r<sub>c</sub> = S<sub>c</sub>/K+V<sub>c</sub> Average rate of profit</td><td>15.7%</td><td>16.8%</td><td>17.7%</td><td>18.4%</td><td>19.0%</td><td>19.4%</td><td>19.7%</td><td>19.8%</td><td>19.8%</td><td>19.7%</td><td>19.5%</td><td>19.4%</td><td>19.2%</td><td>19.1%</td><td>19.0%</td><td>...</td><td>16.2%</td></tr></tbody></table>
These values, achieved in year 14, are maintained for 10 years, i.e. the average lifespan *t* of fixed capital, until year 23, that is to say until the value of total accumulated fixed capital corresponds to *I*<sub>t</sub>*t*. The rate of profit, after having gone through a maximum of 19.8 % in years 7 and 8, then stabilizes at 16.2 %, or 0.5% more than its initial level.

This new example is instructive, as it shows firstly that a significant increase in the rate of surplus-value, which goes from 55% to 74%, if indeed it translates in the long-term into an equally significant rise in the organic composition of capital *q*<sub>c</sub>, which goes from 2.50 to 3.53, does not translate as one would expect at first, on the basis of the model developed in a situation of simple reproduction with no capitalist consumption, into a fall in the average rate of profit.

This takes place even though, as shown in the table on the previous page, we find ourselves in a situation where the hypothesis according to which *t* = 10 has been maintained, from which it follows that *c* > 1/(1+t), as well in year 0, where *c* = 60/320 = 0.188 as in year 23, where *c* = 130/600 = 0.277. If we were in a situation of simple reproduction, then function *r*<sub>c</sub> = *f*(*q*<sub>c</sub>) should be decreasing, as we showed in sub-section 3.1. of chapter 20, and one would expect to record a decrease in the rate of profit. On the contrary, there is a slight increase in the average profit rate at the end of the period, from 15.7% to 16.2%. This can be explained for reasons related to the nature of function *r*<sub>c</sub> = *f*(*q*<sub>c</sub>).

The first reason is very simple, and comes from the fact that in table 2 in reference coefficient *c* is not a constant, which implies that parameter *γ* in function *r*<sub>c</sub> = *f*(*q*<sub>c</sub>) is not a constant either. From then on we have thus different functions giving the rate of profit, one for each value of *c* and therefore of *γ*. This can be verified for example for year 0 and year 23.

In year 0 we have: $\gamma = \frac{c}{1-c} = \frac{0,231}{1-0,231} = 0,30$, which gives us:

$$r_c = \frac{q_c + \gamma t}{(q_c + 1)t} = \frac{q_c + 0,30.10}{(q_c + 1).10} = \frac{q_c + 3,0}{10q_c + 10,0}$$

Whereas at the end of the period we have: $\gamma = \frac{c}{1-c} = \frac{0,277}{1-0,277} = 0,383$, which gives us:

$$r_c = \frac{q_c + \gamma t}{(q_c + 1)t} = \frac{q_c + 0,383.10}{(q_c + 1).10} = \frac{q_c + 3,83}{10q_c + 10}$$

From this it follows that in fact the values of *c* and *γ* change each year between years 1 to 14, and the corresponding function with this *γ* coefficient changes therefore similarly. It can thus be seen that $r_c = f(q_c)$ is not linked by a function of a single variable to the organic
composition of capital, but also depends on the evolution of parameter $\gamma$, which itself expresses the variations in capitalist consumption relative to that of workers. As an example, we have drawn the two curves corresponding to years 0 and 23, which appear in figure 22.1.

<figure><img src="image_19.png" /><figcaption>Figure 22.1 - Curves representing function $r_c = \frac{q_c + \gamma t}{(q_c + 1)t}$ in year 0 and year 23</figcaption></figure>

Moreover, as in the more simple case discussed in section 3 above, where the consumption of capitalists did not appear, the function linking the rate of profit and the organic composition of capital has been defined on the basis of a relationship between the rate of surplus value, which is now $k_c$, and the organic composition of capital, which is now $q_c$. It is relatively simple, as given by equation (13) in chapter 20, that is $q_c = (k_c - \gamma)t$.

However, a second reason explaining that function $r_c = \frac{q_c + \gamma t}{(q_c + 1)t}$ cannot any more explain the evolution of the rate of profit from one year to another is similarly that $t$ is no longer a simple scale factor that does not affect the variation of the function, since one does not instantly switch from the situation of year 0 to the situation of year 23. The above relation between $q_c$ and $k_c$ is thus verified in year 0 and in year 23, but in the interval, and even if $t$ remains constant, which is the case in our example where $t$ remains permanently equal to 10, this relation is not verified, and $q_c \neq (k_c - \gamma)t$. This is so for the same reason as before: in a situation of expanded reproduction, the stock of capital at a given instant necessarily depends on all the previous evolution of the accumulation during the $t-1$ preceding years.

The form of function $r_c = f(q_c)$ shows us nevertheless that it is an increasing function of variable $\gamma$, which is itself an increasing function of variable $c$. The relative increase in
capitalist consumption goes therefore in the same direction as an increase in the rate of profit, which corresponds to intuition. In this numerical example, this increase in capitalist consumption is enough to counterbalance the decreasing effect on the rate of profit of the significant increase in the organic composition of capital. We can also verify from this numerical example that if at the end of the period parameter $c$ remained at its initial level during year 0, i.e. $c = 0.231$ corresponding to $\gamma = 0.30$, but with the same increase in the organic composition, from 2.50 to 3.83, the rate of profit would be 14.1 %, instead of 16.2 %, and would therefore be significantly reduced, also from its initial value of 15.7%. The growth of $c$ thus has the effect of moving $r_c$ curves up and to the right, and this is what explains ultimately the rise in the rate of profit, in spite of the increase in the organic composition of capital.

This shows one more, if it was necessary, the paramount importance of the consumption of capitalists as a driving variable of the economic system.

Incidentally, it is also interesting to note that the rise in the profit rate during the whole 23 year period, even though it is quite limited: from 15.7% to 16.2%, is accompanied by a drop in what we might call the efficiency of capital, rather than the productivity of capital, i.e. ratio $Y / K$. In year 0 we have indeed $Y_0 / K_0 = 310/500 = 0.62$, while at the end of the period we have $Y_{23} / K_{23} = 590/1200 = 0.49$, which represents a 21% decrease in the ratio between the product and the stock of fixed capital. This divergent evolution shows once more that the rate of profit has nothing to do with any productivity of capital, since, as this example shows, the evolution of the efficiency of capital can very well go in the opposite direction from that of the rate of profit.

## 4. The influence of the lifespan of fixed capital and $k_c$

Before concluding this chapter, another question may be worth analyzing, which is the influence that changes in the lifespan of fixed capital may have on the evolution of the rate of profit. In order to do that, we need to go back to equation (22) in chapter 20, namely:

$$r_c = \frac{k_c}{(k_c - \gamma)t + 1}$$

But now we must interest ourselves to the nature of this function as $r_c = f(t)$. Its derivative is: $$\frac{dr}{dt} = \frac{-k_c(k_c - \gamma)}{[(k_c - \gamma)t + 1]^2}$$, which has the same sign as its numerator.

Since we know that by definition $k_c = \frac{k+c}{1-c} = \frac{k}{1-c} + \frac{c}{1-c} = \frac{k}{1-c} + \gamma$, with $k > 0$ and $0 < c < 1$, it follows that we always have $k_c > \gamma$, which implies that $\forall k_c, \forall \gamma, k_c - \gamma > 0$.
Therefore the sign of the derivative $\frac{dr}{dt}$ is always negative, which shows clearly that the rate of profit $r_c = f(t)$ is necessarily a decreasing function of the lifespan of fixed capital, any increase in its duration leading to a fall in the rate of profit, because of the rise that it implies in the organic composition of capital $q_c$, obviously for a given value of $k_c$ and $\gamma$.

As for the effect on the rate of profit of a simultaneous increase in the rate of surplus-value, $k_c$, and the lifespan of fixed capital, assuming now that $r_c = f(k_c, t)$, it is given by the following formula:

$$dr_c = \frac{\delta r_c}{\delta k_c} \cdot dk_c + \frac{\delta r_c}{\delta t} \cdot dt \quad (1)$$

As a result, assuming that $\gamma$ = constant, we get:

$$dr_c = \frac{1}{[(k_c - \gamma)t + 1]^2} \cdot dk_c - \frac{k_c(k_c - \gamma)}{[(k_c - \gamma)t + 1]^2} \cdot dt \quad (2)$$

From this last equation we can infer that to have a positive evolution of the profit rate, such that $dr_c > 0$ implies that:

$$\frac{1}{[(k_c - \gamma)t + 1]^2} \cdot dk_c > \frac{-k_c(k_c - \gamma)}{[(k_c - \gamma)t + 1]^2} \cdot dt \Rightarrow \frac{dk_c}{k_c} > -(k_c - \gamma)dt \quad (3)$$

$$\text{or } \frac{dk_c}{k_c} > (\gamma - k_c)dt \quad (4)$$

From sub-section 3.3. in chapter 20, we know that with the Marxist definition of the profit rate, this rate is a decreasing function of the rate of surplus-value $k_c$ if $c > \frac{1}{1+t}$. Assuming that we are in such a situation (with for instance $t=10$ and $c > 0.091$), then a decrease in $k_c$, which must cause a rise in the rate of profit, can be compensated by a reduction in the lifespan of fixed capital, so that we have:

$$\frac{dk_c}{k_c} = (\gamma - k_c)dt \Rightarrow dt = \frac{dkc}{kc(\gamma - k_c)} \quad (5)$$

For instance an increase in $k_c$ of 10% (from 0.50 to 0.55), coming from an increase in $k$ (the primary rate of surplus-value), since $c$ and $\gamma$ remain constant, would normally imply a reduction in the rate of profit. But this reduction can be compensated by a reduction of the lifespan of fixed capital, given by this last formula, resulting in an offsetting increase in the rate of profit,
Indeed taking $c = 0.20$, $k_c = 0.50$ and $t = 10$, we have
$$r_c = \frac{0.5}{(0.5-0.25)10+1} = 0.1429$$

Whereas with $\frac{dk_c}{k_c} = 10\% = 0.10 \Rightarrow k_c = 0.55$, we have:

$$\begin{aligned} \frac{dk_c}{k_c} &= 0.10 = (\gamma - k_c)dt \\ \Rightarrow dt &= \frac{0.10}{\gamma - k_c} = \frac{0.10}{0.25 - 0.50} = \frac{0.10}{-0.25} = -0.40 \end{aligned}$$

This means that a 10% increase in $k_c$ can be compensated by a reduction in $t$ from 10 years to 9.6 years. If it is the case, we have indeed:

$$r_c = \frac{k_c}{(k_c - \gamma)t + 1} = \frac{0.55}{(0.55 - 0.25)9.6 + 1} = \frac{0.55}{2.88 + 1} = 0.1418$$

The slight difference in both values of $r_c$ comes from the fact that $\frac{dk_c}{k_c}$ is not an infinitesimal amount.

# 5. The realization of the product in the context of extended reproduction with capitalist consumption

## 5.1. Marx's extended reproduction schemes in volume II of Capital

In part III of chapter 21 of volume II of Capital, Marx presents a number of reproduction schemes, that are all of the same form as what he calls an “Initial Scheme for Reproduction on an Extended Scale”, and which is as follows:

$$\text{I. } 4,000c + 1,000v + 1,000s = 6,000$$

$$\text{II. } 1,500c + 750v + 750s = 3,000$$

$$\text{Total or annual social product} = 9,000$$

Then he explains that after one year of capital accumulation we can have a second scheme, where capital accumulation has taken place, and such as:

$$\text{I. } 4,400c + 1,100v + 1,100s = 6,600$$

$$\text{II. } 1,600c + 800v + 800v = 3,200$$

$$\text{Total or annual social product} = 9,800$$
What Marx intends to do is to show that the realization (sale) of what he calls the annual social product is possible even in a situation of extended reproduction, when this social product grows from one period to the other due to the accumulation of additional capital (for an amount of 500 in his example). To some extent he succeeds in his endeavor, by bringing into the picture an appropriate amount of money supply, coming from exports, gold production, or previous hoarding, not to mention credit money. Nevertheless all his schemes are logically flawed, for several reasons, which we will mention briefly:

1) The schemes make no distinction between values and prices

With the reproduction schemes we are at the level of circulation, or realization (sale) of the commodities which have been produced. This is the condition for the economic system to be able to reproduce itself. We are thus in a sphere where the corresponding exchanges (purchases/sales) take place through a transfer of money against commodities, for amounts which are monetary prices. Marx is well aware of that, since as soon as the first chapter of volume 2 and in several other chapters he gives numerous examples in terms of £ (British pounds). But all his reasoning is made in terms of values, as if exchanges took place directly at levels corresponding to the values of commodities. This should not be a surprise, because at this stage Marx has not yet elucidated the question of transformation of values into prices, a question that he addresses only in volume III, published by Engels nine years later. And we know that the methodology that he uses to try and solve this problem is itself a wrong one.

In any case, this confusion between values and prices forbids him to understand that money prices of commodities are always necessarily higher than their money values, which is a condition for surplus-value to be extracted under the form of profits in section II (even with no consumption of capitalists) and for surplus-value to be redistributed among capitalists in section I.

2) Constant capital as Marx defines it, i.e., the total value of raw materials and fixed means of production used up in an accounting period, is a flawed concept

Indeed raw materials are by definition intermediate commodities which are always produced in order to be used in the production process, where they are transformed to be incorporated into other intermediate goods, and ultimately into final goods. As such they never appear on the market for final goods, where they are to be bought, in order to be consumed by workers or capitalists - for consumption goods, or to be invested by capitalists - in the case of fixed capital. Therefore they should not be put in the same category (constant capital) as fixed capital, which is made of final goods.

However this is not what Marx does, because he considers that fixed capital, like intermediate goods, transfers a part of its value to the value of the final product. This is why he puts this part into the same category as raw materials, and treats it as a particular kind of intermediate good.
In any case the values of intermediate goods must be calculated first, but within Marx's conceptual framework constant capital produced by section I would include both intermediate goods which enter into the production of other intermediate goods, on the one hand, and constant capital which enters only into the production and value of final goods, on the other hand.

This would lead to a problem for Marx's schemes, because the annual social product would not be composed only of final goods, as it should be, but also of intermediate goods which do not enter into the production of final goods: i.e. those which enter only in the production of other intermediate goods, and as such do not go to the market. This might be more easily understood if we thought of the process of production as fully integrated, with only one huge firm in each section, producing not only the final goods but also all of the intermediate goods necessary for its final production. Then it would be impossible to wrongly count intermediate goods entering only in the production of other intermediate goods as a part of constant capital, and as a part of the annual social product, which would come down to counting them twice. Only intermediate goods entering in the production of final goods would be accounted for. This leads us to a third and final reason for which the extended reproduction schemes in fact cannot teach us anything about the true nature of the reproduction process.

3) The wrong assumption that fixed capital transmits its value to the product.

Indeed with Marx's assumption above, as we already showed, a part of fixed capital would also wrongly be accounted for twice in his reproduction schemes: first as a final good sold on the market of final products, and second for the amount supposed to transmit its value to the final goods into which this lost value would be "incorporated". In fact once whatever good, including fixed capital, has been sold on the market for final products against an income which is transferred from the buyer to the seller, this good cannot be sold a second time as a final good. If it is sold again, it will necessarily, for logical reasons, be sold as a second-hand good, and as such will not create any new income: what is spent by the buyer is obtained by the seller, which means that from a macroeconomic point of view no additional income is created by these transactions concerning goods previously produced and exchanged.

Because of these various flaws there was no chance that Marx's reproduction schemes could be correct, which leads us to propose now what we deem is a right approach to this question.

## 5.2. Another way to deal with reproduction schemes

At the outset, it should be emphasized that the realization of the product is a precondition for its reproduction. This explains that our schemes for realization are the basis for understanding reproduction problems. In the case of simple reproduction the product and its structure are identical from one period to the other, which means that the value of the product does not change, both globally and in each of the two sections of the productive system, and therefore that $L_I, L_{II}$ and by definition $k$, which connects them (since $k = \frac{L_I}{L_{II}}$) are constant.
If we want to compare our schemes with Marx's schemes, since they involve circulating capital, as a part of constant capital, the first thing to do is to define correctly the values of both intermediate commodities and final commodities, which has been done already in chapter 6 of the present study. Indeed we showed in this chapter that values could be calculated with a first method as given by the following simplified system of equations:

$$V_1 = V_1 A_{11} + L_1, \text{ for intermediate commodities} \quad (6)$$

$$V_2 = V_1 A_{12} + L_2, \text{ for final commodities} \quad (7)$$

This system could be solved as:

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

$$V_2 = L_1 (I - A_{11})^{-1} A_{12} + L_2 \quad (9)$$

We showed also that values could be calculated with a second method as given by another system of equations:

$$V_1 = L_1 X_{11} \quad (10)$$

$$V_2 = L_1 X_{12} + L_2 \quad (11)$$

Finally, we pointed out that both systems were equivalent, because we had:

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

$$X_{12} = (I - A_{11})^{-1} A_{12} \quad (13)$$

So that we could replace $X_{11}$ and $X_{12}$ in equations (10) and (11) by their values given by equations (12) and (13) to obtain equations (8) and (9).

This allowed us to show finally that the value of the product is equivalent to total direct labor time. Indeed, calling $y_1$ and $y_2$ the column vectors of the output of intermediate commodities and final commodities, respectively, from equations (8) and (9) we could write that the value of the total product is equal to:

$$V_1 y_1 + V_2 y_2 = L_1 (I - A_{11})^{-1} y_1 + L_1 (I - A_{11})^{-1} A_{12} y_2 + L_2 y_2 \quad (14)$$

This, from equations (10) and (11), is equivalent to:

$$V_1 y_1 + V_2 y_2 = L_1 X_{11} y_1 + L_1 X_{12} y_2 + L_2 y_2 \quad (15)$$

This last equation gives us indeed the overall quantity of direct labor time used in the production process, first for the production of intermediate commodities, i.e. $L_1 X_{11} y_1 + L_1 X_{12} y_2$, and second for the production of final commodities, i.e. $L_2 y_2$.
Going back now to reproduction schemes, it is clear that to establish them we need to make a change in the system of references, because we are dealing here with final incomes, which are spent only to buy final commodities. At this stage, money wages have been paid to workers, and we are in the theoretical realm of money values. Equation (15) shows that all the money value of the product can be considered as equivalent to the total wages paid to all workers. Moreover, since wages and the transfer incomes deriving from them are spent on the markets for final goods, all the transactions to buy and sell intermediate goods are supposed to have taken place previously, during the production process (otherwise production could not have been carried out up to its final stage).

Moreover, the quantity of direct labor time must be distributed no more between the production of intermediate goods and the production of final goods, but between the production of consumption goods and the production of fixed capital goods, this production being integrated for each of them, with section I producing both its own intermediate goods and fixed capital and section II producing its own intermediate goods and consumption goods.

This implies first that the first two terms of equation (15) corresponding to intermediate goods must be split between these two sections, in such a way that the total labor of the system is spent indirectly and directly in each of these two sections. Thus the overall quantity of direct labor time $L_1 X_{11} y_1 + L_1 X_{12} y_2 + L_2 y_2$ is split between the labor time used indirectly and directly in the production of consumption goods, or $L_{II}$, and the labor time used indirectly and directly in the production of fixed capital, or $L_I$. This implies as a second step that this equation be monetized, which is done by multiplying the quantities of labor by the average money wage per unit of time $\bar{w}$. By taking it also as unit so that $\bar{w} = 1$, we get:

$$L_{II} = \bar{w} \quad L_{II} = L_{II} \text{, equivalent to the money value of consumption goods.}$$

$$L_I = \bar{w} \quad L_I = L_I \text{ equivalent to the money value of fixed capital goods.}$$

Reproduction implies also that exchanges take place at monetary prices which must allow first for the creation or extraction of surplus-value $S$. This process, as we have seen, is carried out in section II through the realization of consumption goods and implies that their overall price $P_{II}$ must differ from their overall money value $L_{II}$, and be higher than this value in such a way that this overall price is:

$$Y_{II} = P_{II} = L_{II} + S = L_{II} + L_I + cL_{II} (1 + k_c) = L_{II} (1 + k_c) \quad (16)$$

This equation shows that this price is the sum of wages in section II ($L_{II}=W_{II}$) plus surplus-value $S$. The realization (or sale) of the product is thus carried out at a price which is the sum of wages in both sections ($L_I+L_{II}=W_I+W_{II}$) spent on consumption goods, plus the consumption of capitalists, as a share $c$ of total consumption, measured by the total price of these goods. This share is the same if measured in terms of values, since in this section prices are globally proportional to values.