The growthmodel of Marx according to Sam de Wolff

First insertion on Heterodoxe Gazet Sam de Wolff: 6 february 2012

In a preceding column the dynamical growthmodel of Karl Marx has been described. Marx uses his model especially in order to study whether the various industrial branches can trade with each other, without causing economic disturbances. In the twentieth century authoritative economists have elaborated on this concept. Worth mentioning are M. kalecki and J.M. Keynes in the thirties, and especially P. Sraffa in the sixties. Several years before, namely in chapter 9 of his main work Het Economisch Getij (The economic tide)¹ Sam de Wolff has analysed the growthmodel of Marx. De Wolff performs his study notably in order to refute the statement of Rosa Luxemburg, that the capitalism is principally unstable. In her publication Die Accumulation des Kapitals she alleges that in the capitalist system the consumptive demand is structurally insufficient. This Krise in Permanenz can only be alleviated temporarily, namely as long as the capitalist system can export its products to pre-capitalist societies, like at the time the colonies.

De Wolff does not share the standpoint of Luxemburg. In order to prove his right he applies the two-sector model, already introduced in the preceding column, which distinguishes between the generation of the means of production (deparment I) and the generation of the consumer goods (department II). He brings in a refinement, by splitting the department II in two subdepartments: the subdepartment IIa for the generation of the wage goods for the workers, and the subdepartment IIb for the generation of the luxury goods for the entrepreneurs. This subdivision is not absolutely necessary for his polemic with Luxemburg. However further on in his analysis he will use the department IIb in order to study the influence on the economy of the money, in the form of gold reserves.

So de Wolff summarizes the economic system in the following three formulas:

(1)     K_I + L_I + M_I = X_I
(2)     K_IIa + L_IIa + M_IIa = X_IIa
(3)     K_IIb + L_IIb + M_IIb = X_IIb

This shows how each department produces a certain amount of product X. In the three-department model of De Wolff these products are the means of production X_I, the wage goods X_IIa, and the luxury goods X_IIb. In principle these quantities can be expressed in for instance the physical amounts of product, But Marx prefers to use the amount of expended labour time (in years, or days, or hours) as a measure. Of course the expended labour time contributes to the value of the product. For Marx the unit of labour time is the universal measure of value, and as such it is leading for its money value. The labour value of the total product must be divided over the various factors, that are involved in the production process. The other symbols in the formulas 1, 2 and 3 have the following meaning:

• K represents the value of the means of production and of the raw materials. These are consumed during the production process, either as material which merges into the final product, or as substance that is necessary for the production process (for instance energy), or as wear of the machinery (tools, buildings and the like). Of course these means of production have to be bought before the start of the production. So the sum K has to be advanced by the entrepreneurs.
• L represents the value of the wage fund for the workers. Marx assumes that the wages are paid at the start of the production process, and thus must be advanced as well by the entrepreneur. The wage fund determines the production costs, together with K.
• M is the surplus value. During the production process this part of the production value is added to the product as an extra², and it is appropriated by the entrepreneurs as their reward. They must hand over a part of their surplus value to the state, in the form of taxation. The state uses its yield of taxation for bearing the costs of the collective services. Here these are the consumptive expenses, like education, health care and defence, that are not paid out of the nett wage sum. The collective services are irrelevant for the present argument, en can be left out of consideration. Apart from this part the entrepreneurs are free to choose how to expend their surplus value.

In the preceding column a formula has been derived, that descrives the exchange between the departments (see the formula 6 there). However this formule does not hold for the calculations by De Wolff, due to the spitting of the department II. Besides De Wolff completely abandons the idea of proportional growth. For he adheres to the vision of Marx, that the entrepreneurs try to improve the productivity of labour. For in this way they can keep the wage fund at a minimum. However this implies a continuous automation, and as a consequence the material costs K will rise faster than the wage costs. The ration K/L, which is called by Marx and also by De Wolff the organic composition of the production, exhibits a rising trend³. The result of this choice is that the technical coefficients in the growthmodel all become time dependent. In this way they become less relevant, and can not be used to advantage, like in the method of the preceding column about the growthmodel.

In a model that assumes a constant change of production techniques the growth will usually not be proportional. Each department i has her own growth factor, defined by the formula:

(4)     G_i(t) = X_i(t+Δt) / X_i(t)

In the formula 4 G_i(t) represents the growth in the time interval Δt that follows the time t. Each time interval Δt is commonly called a period. Suppose that the first period starts at t=0. In his book De Wolff presents a numerical example⁴, and therein chooses G_I = 1.1 and G_IIa = 1.05 for all periods. This implies that the department I will become ever more dominant. Of course this can also be expected because of the assumed trend towards an unlimited automation. Besides De Wolff puts the so-called rate of surplus value m' equal to 1 for all periods and for all departments: M_I/L_I = M_IIa/L_IIa = M_IIb/L_IIb = 1. The uniform rate of surplus value M/L for all departments in the system is a distinguishing feature for the economic theory of Marx. It means, that the workers in each industrial branch take care to moderate their production of surplus value, in proportion to their wage, in accordance with the results of the other branches. Actually the stability of m' does not go well with the view of Marx. This contradiction will be addressed again at the end of this column.

De Wolff goes on to fix the organic compositions for the first period on K_I/L_I=3, K_IIa/L_IIa=2 and K_IIb/L_IIb=1. So here the generation of the means of production is most capital intensive, which corresponds with reality. He asks himself how the economic proportions should be in order to satisfy all these demands. There are twelve unknowns, namely K, L, M and X in the three departments, all of course with regard to the period 1. The requirements for the rate of surplus value and for the organic composition yield six equations. In addition there are the three formulas 1, 2 and 3. And finally because of the formula 4 there are the two growth equations:

(5)     X_I(1) = G_I(0) × X_I(0) = G_I(0) × (K_I(1) + K_IIa(1) + K_IIb(1))
(6)     X_IIa(1) = G_IIa(0) × X_IIa(0) = G_IIa(0) × (L_I(1) + L_IIa(1) + L_IIb(1))

The formulas 5 and 6 express that at the start of the period 1 the total amount of capital goods K is the yield of the department I during the period 0, and likewise that at the start of the period 1 the total amount of wage goods L is the yield of the department IIa during the period 0. If desired the formulas 5 and 6 can be combined with the formulas 1 and 2 to

(7)     L_I(1) + M_I(1) = ΔK_I(1) + G_I(0) × (K_IIa(1) + K_IIb(1))
(8)     K_IIa(1) + M_IIa(1) = ΔL_IIa(1) + G_IIa(0) × (L_I(1) + L_IIb(1))

In the formulas 7 and 8 ΔK_I(1) and ΔL_IIa(1) are respectively the extra produced amounts of capital- and wage-goods in the period 1. They form the accumulation during the period 1, which guarantees that in the period 2 there is a sufficient number of goods in behalf of the nett investments and of the expansion. These formulas are the equivalent of the exchange formule 6 from the preceding column about the dynamic growthmodel of Marx, albeit now without the technical coefficients. It does not matter, whether the growth formulas 5 and 6 are used, or the exchange formules 7 and 8, because in each case there are now eleven equations with twelve unknowns. Apparently one quantity can be chosen at will, and De Wolff chooses the identity K_I(1)=3333, because with this value all other quantities turn out to be (almost) natural numbers. The resulting solution of the now twelve equations is shown in the table 1.

The table 1 shows that the period 1 differs from the preceding period by additional means of production to the value of ΔK=505, and by additional wage-goods to the value of ΔL=112. In the next period the entrepreneurs will add these products to the production process, and therefore they represent an accumulation to the value of 617. The accumulation must be paid from the surplus value, so that the entrepreneurs have left only the value 1623 for their consumption of luxury goods. The department IIB does not accumulate, because the luxury goods do not contribute to the production process. The production factors are generated completely by the departments I and IIa. The accumulation quote⁵ for the entire economy is a=ΔX/M=0.2754. The date in the table 1 do not allow to calculate the accumulation quote in each of the departments. In other words, the distribution of the accumulated value of 617 over the departments is yet unknown.

The distribution of the accumulated value over the departments determines how the economic structure in the period 2 will look like. De Wolff has already guided the development by assuming that the growth factors and the rate of surplus value are independent of time. In his book he takes on the task to study the consequences for the period 2. Here he makes the surprising decision not to study the quantities K, L, M and X themselves, but their increases⁶, namely their accumulations ΔK, ΔL and ΔM. The accumulation ΔX is already known from the table 1. Because of the three departments, there are nine unknown quantities, which requires nine equations to calculate them. De Wolff sets out to derive this set of equations. Since the rate of surplus value is independent of time, its increases must have m'=1 as well. So ΔM_I/ΔL_I = ΔM_IIa/ΔL_IIa = ΔM_IIb/ΔL_IIb = 1.

De Wolff defines the total accumulation in the economic system as follows

(9)     ΔX_I(t) = ΔK_I(t) + ΔK_IIa(t) + ΔK_IIb(t)
(10)     ΔX_IIa(t) = ΔL_I(t) + ΔL_IIa(t) + ΔL_IIb(t)

Together with the growth formula 4 these two formulas yield the result

(11)     ΔX_I(2) = X_I(2) - X_I(1) = G_I × (X_I(1) - X_I(0)) = G_I × ΔX_I(1)
(12)     ΔX_IIa(2) = G_IIa × ΔX_IIa(1)

So the total accumulation of the means of production and of the wage goods exhibit the same growth as the supplies X_I and X_IIa themselves. Hence ΔX_I(2) and ΔX_IIa(2) are known quantities, for ΔX_I(1) and ΔX_IIa(1) can be read from the table 1. The result is ΔX_I(2)=555 and ΔX_IIa(2)=118. In this way again two formulas are found

(13)     555 = ΔX_I(2)= ΔK_I(1) + ΔL_I(1) + ΔM_I(1)
(14)     118 = ΔX_IIa(2)= ΔK_IIa(1) + ΔL_IIa(1) + ΔM_IIa(1)

Togeteher with the three formulas for the rate of surplus value the formulas 9, 10, 13 and 14 form a set of 7 equations. Two additional formulas are needed to complete the set. These two formulas concern the growth rate of the organic composition in the departments, and can be chosen at will. Table 1 shows that in the period 1 the identities are K_I/L_I=3 and K_IIa/L_IIa=2. The growth of these ratios for the period 2 requires, that ΔK_I/ΔL_I>3 and ΔK_IIa/ΔL_IIa>2. De Wolff chooses for these ratios respectively the values 6 and 2.5. Now the set is complete and can be solved. The result⁷ is shown in the table 2, and the corresponding economic structure in the period 2 is given in the table 3. Besides table 2 mentions the accumulation quotes for each department, a_K(1) = ΔK(1)/M(1) and a_L(1) = ΔL(1)/M(1), which determine the distribution of the surplus value at the end of the period 1. Since the growth in the department IIb is consumed completely, the quantity ΔX_IIb in the table 3 (there interpreted as an accumulation quantity) is equated to zero.

It is instructive to examine in imitation of the formulas in the book of Eva Müller the change of the technical coefficients with time. Table 4 shows these coefficients for the periods 1 and 2. In stead of the coefficients α and γ, as used by Müller, here the coefficient μ=M/X is reproduced.

The table 4 shows that in the example of De Wolff none of the coefficients is a constant, contrary to the example of Eva Müller. Besides the techniques shift towards a higher capital intensity, which augments κ and reduces λ, especially in the department I. De Wolff has clearly modelled a more complex economic system than Müller. His model is more advanced. However just like the model of Müller it has the disadvantage that it does not lead tot a uniform average profitrate r. This is illustrated in the table 5, where the profit rates in the three departments are shown for the periods 1 and 2. According to Marx unequal profit rates in the various departments are not possible, in the long rung. For it is obvious that for example in the situation of table 5 all entrepreneurs will relocate their activities to the very profitable department IIb. However, Marx did not succeed in developping a model that guarantees a uniform profit rate in all departments. Also De Wolff does not have a solution for this problem. This undermines somewhat the application in practice of his argument⁸.

In any case De Wolff has proven here, that an increasing capital intensity is in itself not a disturbance of the production process. Nevertheless, he acknowledges, that his assumption of a time-independent rate of surplus value is a bit dubious⁹. For the entrepreneurs carry through the automation, just because they want to control the wage sum. The automation increases the labour productivity, so that the wage goods will become cheaper. Therefore the nominal wage (in money value) can fall, while still the real (fysical) wage is preserved, or may even rise somewhat. Then the entrepreneurs will have left a larger profit. In other words, the rate of surplus value will rise in principle due to the increasing capital intensity. Since De Wolff has omitted this trend from his model, it can not be used to conclude definitely, that Rosa Luxemburg was wrong with her hypothesis of a Krise in Permanenz¹⁰.

1. See Het Economisch Getij (1929, J. Emmering). (back)
2. In our times it is usually assumed, that the wage fund is paid only after the termination of the production process. Then it is not a part of the advance, but it is generated during the production process itself. In this case the wage fund becomes a part of the added value (surplus value). (back)
3. In our days it would be said, that the capital intensity of the production shows a continuous increase. Note however, that the K of Marx also includes the raw materials and other material resources. In modern language these do not belong to the capital investments, because they return immediately at the sale of the products. (back)
4. See p.385 and following in Het Economisch Getij. (back)
5. In the modern economic terminology it is common to use the investment quote as a measure. She is the ratio between the gross investments and the total product. In our days the advances for wages and for the raw materials and resources are not included in the investments, contrary to the habit of Marx. For these return immediately at the sale, while the investments in machinery (tools, building) must be amortized during many years. (back)
6. Eva Müller suggests on p.233 and p.237 of her book Volkswirtschaftlicher Reproduktionsprozeβ und dynamische Modelle (1973, Verlag Die Wirtschaft) this possibility, but she does not give a work out in formulas. (back)
7. A check of the tables 2 and 3 with the results of De Wolff on p.394 shows that his values have been rounded off to natural numbers in a rather inaccurate way. (back)
8. A solution was only presented in 2007, when Andrew Kliman in his book Reclaiming Marx's "Capital" (Lexington Books) made public his Temporal Single System Interpretation (TSSI). The TSSI of Kliman is a model (he calls it an exegetic interpretation himself, that is to say, a textual explanation of the theory of Marx), thatallows to perform calculations with a universal profit rate r, while still maintaining the undisturbed exchange of products between the departments. (back)
9. See p.397 and further in Het Economisch Getij (back).
10. Still it is remarkable, that De Wolff makes this reservation. Namely he could have chosen a larger rate of surplus value for the period 2, without causing problems in his model. Then the calculation can proceed just like now. Therefore it seems that De Wolff does refute the hypothesis of Luxemburg. (back)