The dynamical growthmodel of Karl Marx

First insertion on Heterodoxe Gazet Sam de Wolff: 2 january 2012

The economic theory of Karl Marx flourished in the period between 1880 and about 1920, when she was the intellectual foundation for the rapidly expanding labour movement. As the labour movement integrated into the existing order, it distanced itself gradually from the marxist doctrine. In Russia, where after a coup d'état the Leninists had established a dictatorship, the theory of Marx became fossilized into a dogma and a state ideology. Between 1966 and 1980 the theory of Marx underwent a miraculous revival¹. This revival was more motivated by culture than a result of new scientific break-throughs. At least two causes can be found for this renewed interest:

• For decades the world had been ravaged by the existence of the Cold War: the peace based on the principle of the Mutually Assured Destruction. In the sixties the peace movement gained the upper hand, which resulted in a new Ost-Politik. The marxist debate was very suitable for starting the dialogue with the eastern rulers.
• The secularisation rapidly gained ground, and the churches lost their credibility. Especially the former catholics saw the philosophy of Marx as an alternative for their own lost conception of the world. Perhaps also they recognized in the central organization structure of the Leninists the culture of their own ecclestiastical organization.

In the philosophical domain the renewed interest in the works of Marx actually led to some advances. However the interest in the economy of Marx only caused the publication of popular writings, that did not pass the repetition of old knowledge. In the Netherlands the Leninist Party was instrumental in this process, because she could use it as a source of propaganda. In scientific circles the economy of Marx had been superseded almost completely by the production model of Sraffa². This current is also called the neoricardian theory. Therefore in the years 1965-1985 the offer of scientific studies of the theory of Marx was completely confined to the research in the Leninist states. There the dynamical growthmodel of Marx had become the foundation fot all kinds of models for economic planning.

In this column the dynamical growthmodel of Marx is explained once more, notably because this model has remained relatively unknown in the west. Marx designates the growth as the expanded reproduction, in order to distinguish it from the simple reproduction without growth. The formalism in this column is copied from a standard textbook, written by the East-German economist Eva Müller³. In principle the growthmodel belongs to the category of interweaving models, which describe the cohesion between all industrial branches. In its most simple form the model of Marx contains two branches, denoted as departments, namely (I) the department for the production of the means of production, and (II) the department for the production of consumer goods. In macroeconomics this is a logical division, which later, halfway the twentieth century, has also been applied by M. Kalecki and J.M. Keynes in their models. For the department I contains exactly the investment goods, which are meant to generate the capital interest, without directly satisfying the consumer needs. Notably the department I serves the satisfaction of human needs.

Each department produces a certain amount of product X. So in the two-department model of Marx these are X_I and X_II. These quantities could be expressed for instance in physical amounts of product, but Marx prefers to use the expended amount of labour time (in years, or days, or hours) as a measure. Of course the labour time works through in the value of the product. For Marx the unity of labour value is the universal expression of value, and as such is leading for its money value. The labour value of the final product must be subdivided among the various actors, that participate in the productions process. Omitting for the moment the index of the department, the value can be expressed concisely in the formula:

K + L + A + C = X     (1)

The symbols in the formula 1 have the following meaning:

• K represents the value of the means of production and of the raw materials. These are used up during the production process, either as material that merges into the product, or as material that is needed during the production process (for instance, energy), or as wear of the equipment (machinery, buildings and the like). Obviously these means of production have to be purchased before the production can begin. So the sum K must be advanced by the entrepreneurs.
• L represents the value of the wage sum of the workers. Marx assumes, that the wages are paid at the beginning of the production process, and thus must be advanced by the entrepreneur as well.
• A is the amount of value, that becomes available for the accumulation. This value is needed for the nett investments in the equipment, and also for the future expansion of the amount of raw material in the production process. It is important to realize, that the future expansion of the production process will increase the wage sum as well. For practical reasons Eva Müller does not include this component into A, but into C.
• C is a rest value, which serves to cover a variety of expenses. She is collected in order to pay for the increase of the wage sum, and besides for the income of the entrepreneurs and for collective provisions. The part of the entrepreneurs also serves to pay for the interest and rent. Here the collective provisions are consumptive expenses, like education, health care and defence, that are not paid from the nett wage sum. Since the collective provisions are usually supplied by the state, that part of C is collected by a taxation on X. In fact the entrepreneurs will first collect C entirely as a profit, and afterwards use it to pay the taxes and the growth of the wage fund. In the socialist economy the entrepreneurs are a part of the state machinery. This explains why Eva Müller considers that C is a useful quantity.

Certainly since the publications of Sraffa is is common to normalize all equations on the amount of product X. Then the formula 1 changes into

κ + λ + α + γ = 1     (2)

The quantities κ, λ, α and γ are called the technical coefficients, because their value is dependent on the chosen production technique. Formally they are defined by κ=K/X, λ=L/X, α=A/X and γ=C/X. This manner of writing is convenient notably for dynamical growth models, because here the product X will increase with time. This implies that also K, L, A and C are dependent on the time t. However if it is assumed that thes four quantities increase proportional with X, then the coefficients κ, λ, α and γ will be time independent constants. The profit rate r and the accumulation quote a can be expressed directly in terms of the technical coefficients. The result is r = (α+γ) / (κ+λ) and a = α / (α+γ).

In this formalism the growth behaviour of the economic system is given by the formula

K(t+Δt) = K(t) + A(t)     (3)

The accumulated value A on time t is used to expand the means of production and the raw materials after the expiration of the period Δt. The formula 3 is simply transposable into a form with coefficients

κ(t+Δt) × X(t+Δt) / X(t) = κ(t) + α(t)     (4)

The ratio X(t+Δt) / X(t) is sometimes called the growth factor and represented by the symbol G(t). When κ is constant with time, than the formula 4 obtains the sumple form:

α = κ × (G - 1)     (5)

After the formulas within a single department have been set up in this way, it can be determined how the transactions between the departments I and II will unfold. On time t the department I generates the means of production, which it needs for itself after a time Δt. In addition it needs to generate the means of production, which the department II will need on time t+Δt. On the other hand the department II must produce on time t the consumer goods, which his workers and those of the department I will need on the time t+Δt. For both the means of production and the consumer goods must be available at the start of the production process. Note here, that in the model of Eva Müller the growth ΔL_I = L_I(t+Δt) - L_I(t) of the wage goods (consumption of the workers) is paid from the yield C_I. Besides the department II should on time t also produce consumer goods for the capital owners in the department I, as well as goods for the taxation in the department I. However these can not be advanced. In other words, the departments I and II must exchange a part of their production in such a way, that K_II(t+Δt) is equal to L_I(t) + C_I(t). In terms of the technical coefficients this becomes

κ_II(t+Δt) × G_II(t) × X_II(t) = (λ_I(t) + γ_I(t)) × X_I(t)     (6)

The formula 6 completes the growthmodel of Marx, and makes ready everything for concrete calculations⁶.

It is instructive to repeat the calculations, that Marx himself uses in volume 2 of his book Das Kapital⁷. The numbers are shown in the table 1. The features of the example become clear quickly, when the quantities are transformed into the technical coefficients. For then it appears, that Marx supposes that for each period the coefficients are constants. Their values are shown in the table 2.

In this case of constant coefficients the calculations of growth are relatively easy. Of course the factor of growth G in both departments is important, and they can be calculated with the formula 5. It appears that G_I and G_II have the same value, namely 1.1. In each period the economy grows with 10%. Such a case implies a proportional growth, because the economy expands without altering the economic structure. The accumulation quotes are a_I=0.4 and a_II=0.2. The profit rates turn out to be r_I=0.2 and r_II=0.3333. It should be noted here, that according to the theory of Marx deviating rates of profit can not last in the different branches. Therefore his example is apparently an academic finger exercise. Incidentally the rates of surplus value are identical in both branches, namely 1.

The economic system can exist only, as long as the exchange condition of formula 6 is satisfied. She dictates that the ratio X_II/X_I must have the value 0.4849. Table 1 shows that this is indeed the case for the period 1. The proportional growth guarantees that this ratio will remain onchanged in the next periods. Of course the situation can be imagined wherein some technical coefficients do become time dependent. Marx gives an example of this, in the pages just mentioned of volume 2 of Das Kapital. Here is is summarized in the table 3.

From the table it is immediately obvious, that in the period 0 the growth factors G_I and G_II are unequal. Now G_II is no longer 1.1, but merely 1.067. This means that the technical coefficients in the period 1 obtain another value than in the period 0, and consequently the formula 5 does not hold. However the formula 4 can be used to calculate α_I(t), and α_II(t) from the known values κ_I(t), G_I(t), κ_II(t) and G_II(t). Note here that κ_I(t) and κ_II(t) remain constant in all periods 0, 1, 2, ... . It follows that α_I(0) = 0.0667 and α_II(0) = 0.0333. A comparison with the table 2 reveals that in the period 0 the same α_I holds, but that α_II is smaller than in the period 1. During the period 0 in the department I a value A_I=400 is accumulated, and in the department II a value A_II=100. From this it follows immediately that during the period 0 γ_I=0.1 and γ_I=0.2167. This lays down completely the technical structure during the period 0.

The complete example of Marx starts apparently with a disproportional growth in the period 0, after which in the next periods the growth becomes proportional. In the period 0 the path of proportional growth could only be reached by momentarily diminishing the growth in the department II below the growth of the department I. An alternative interpretation is that in the long run the consumptive offer can increase by a temporary rise in the growth of the department I. The growth of the department I can be maximal, when the department II does not grow, and therefore thus not require additional machines. Then the department I can keep all of its extra produced machines completely for itself.

1. An arbitrary grab at the abundance of publications: De actualiteit van Marx (1965, Kruseman) edited by G. Harmsen, De jonge Marx (1962, Veenman en Zonen) by B. Delfgaauw, Karl Marx (1975, Uitgeverij Ankh-Hermes) by H. van Praag, or De visie van Marx (1975, Boom) by R.C. Kwant. (back)
2. The standard work of P. Sraffa is Production of Commodities by Means of Commodities (1960, Cambridge University Press). (back)
3. See chapter 9 in Volkswirtschaftlicher Reproduktionsprozeβ und dynamische Modelle (1973, Verlag Die Wirtschaft) by Eva Müller. (back)
4. At present it is usually assumed, that the wage sum is paid only after the production process. Then he is not contained in the advance, but is created during the process itself. Then the wage sum becomes a part of the added value (surplus value). (back)
5. The ratio A/K is called the accumulation rate. In the present economic language the common measure is the investment quote. She is the ratio between the gross investments and the total product. In our days the advance for raw materials and the like is not included in the investments, contrary to Marx. For raw materials and the like are earned back completely right at the sale, while investments in equipment (machinery, buildings) must be depreciated during many years. (back)
6. Perhaps the equilibrium equation 6 is not very transparent for the reader. This is due to the quantity C, that Eva Müller has introduced in her formulas. According to her this is practical, because in the socialist economy C is an important quantity for planning. In fact the original equilibrium equation of Marx is more lucid. Marx states simply, that on time t department I produces all means of production for the next period. Therefore the identity K_I(t+Δt) + K_II(t+Δt) = X_I(t) = K_I(t) + L_I(t) + M_I(t) holds. In other words, K_II(t+Δt) = L_I(t) + M_I(t) - ΔK_I(t). Note that ΔK_I(t) is simply another way of describing the value of accumulation A_I(t). In this way Marx finds the equilibrium equation K_II(t+Δt) = L_I(t) + M_I(t) × (1-a_I(t)), in which a is the accumulation quote. This formula is in essence the samen as the formula 6, and therefore can also be used for the calculations with the technical coefficients. Then the quantity C does not enter the picture. For those who like to follow in the footsteps of Marx, it should be remarked that he actually writes the exchange formula in the form K_II(t+Δt) = L_I(t+Δt) + M_I(t) - (ΔK_I(t) + ΔL_I(t)). This can be rewritten as K_II(t+Δt) = L_I(t+Δt) + M_I(t) × (1-b_I(t)). In other words, Marx actually uses the accumulation quote b(t), which represents the accumulation in both the means of production and in the wage sum. Marx treats the means of production and the wages on an equal footing, because the both are a part of the production costs. (back)
7. See p.505 of Das Kapital volume 2 (1974, Dietz Verlag) by K.H. Marx en F. Engels. (back)