Logical and mathematical basis of the law of breakdown

Apart from the arithmetical and logical proofs that we have been given already, mathematicians may prefer the following more general form of presentation which avoids the purely arbitrary values of a concrete numerical example.

Meaning of the symbols

c = constant capital. Initial value = c_o. Value after j years = c_j
v = variable capital. Initial value = v_o. Value after j years = v_j
s = rate of surplus value (written as a percentage of v)
a_c = rate of accumulation of constant capital c
a_v = rate of accumulation of variable capital v
k = consumption share of capitalists
S = mass of surplus value, being:

${\displaystyle k+{a_{c}\cdot c \over 100}+{a_{v}\cdot v \over 100}}$

Ω = organic composition of capital, or c:v
(Correction with respect to Grossman's text: From the formula below it follows that Grossman means by Ω the initial value of the organic composition of capital ${\displaystyle c_{0}}$:${\displaystyle v_{0}}$)
j = number of years

Further, let
${\displaystyle r=1+{a_{c} \over 100}}$

and let
${\displaystyle w=1+{a_{v} \over 100}}$

Formula

After j years at the assumed rate of accumulation a_c, the constant capital c reaches the level: ${\displaystyle c_{j}=c_{o}\cdot r^{j}}$

At the assumed rate of accumulation a_v, the variable capital v reaches the level: ${\displaystyle v_{j}=v_{o}\cdot w^{j}}$

The year after (j + 1) accumulation is continued as usual according to the formula: ${\displaystyle S=k+{c_{o}\cdot r^{j}\cdot a_{c} \over 100}+{v_{o}\cdot w^{j}\cdot a_{v} \over 100}={s\cdot v_{o}\cdot w^{j} \over 100}}$

whence

${\displaystyle k={v_{o}\cdot w^{j}(s-a_{v}) \over 100}-{c_{o}\cdot r^{j}\cdot a_{c} \over 100}}$

For k to be greater than 0, it is necessary that: ${\displaystyle {v_{o}\cdot w^{j}(s-a_{v}) \over 100}>{c_{o}\cdot r^{j}\cdot a_{c} \over 100}}$

k = 0 for a year n, if: ${\displaystyle {v_{o}\cdot w^{n}(s-a_{v}) \over 100}={c_{o}\cdot r^{n}\cdot a_{c} \over 100}}$

Note this line follows the German original in Das Akkumulations- und Zusammenbruchsgesetz des kapitalistischen Systems (zugleich eine Krisentheorie) because it is misspecified in the condensed English translation.

The timing of the absolute crisis is given by the point at which the consumption share of the entrepreneur vanishes completely, long after it has already started to decline. This means:

${\displaystyle ({r \over w})^{n}={s-a_{v} \over \Omega \cdot a_{c}}}$

whence n = ${\displaystyle {log\left({\frac {s-a_{v}}{\Omega \cdot a_{c}}}\right)} \over {log\left({\frac {100+a_{c}}{100+a_{v}}}\right)}}$

This is a real number as long as s > a_v

However, this is what we assume anyway throughout our investigation. Starting from time-point n, the mass of surplus value S is not sufficient to ensure the valorisation of c and v under the conditions postulated.

Discussion of the formula

The number of years n down to the absolute crisis thus depends on four conditions:

1. The level of organic composition Ω. The higher this is the smaller the number of years. The crisis is accelerated.
2. The rate of accumulation of the constant capital a_c, which works in the same direction as the level of the organic composition of capital.
3. The rate of accumulation of the variable capital a_v, which can work in either direction, sharpening the crisis or defusing it, and whose impact is therefore ambivalent.
4. The level of the rate of surplus value s, which has a defusing impact; that is, the greater is s, the greater is the number of years n, so that the breakdown tendency is postponed.

The accumulation process could be continued if the earlier assumptions were modified:

1. The rate of accumulation of the constant capital a_c is reduced and the tempo of accumulation slowed down.
2. The constant capital is devalued which again reduces the rate of accumulation a_c.
3. Labour power is devalued, hence wages cut, so that the rate of accumulation of variable capital a_v is reduced and the rate of surplus value s is enhanced.
4. Finally, capital is exported, so that again the rate of accumulation a_c is reduced.

These four major cases allow us to deduce all the variations that are actually to be found in reality and which impart to the capitalist mode of production a certain elasticity ...

Much of the remainder of Grossman's book is devoted to exploring these "elasticities" or counter-crisis tendencies, tracking both their logical and their actual, historical development. Examples of each would include:

1. Depressed interest rates, investment capital transferred to unproductive speculation, e.g. housing stock, art objects.
2. Enlarged state sector bleeds value from the accumulation process via taxes. Wars destroy capital values.
3. The reserve army of labour (unemployed) created to discipline wage claims.
4. Imperialism