A Mathematical Formulation of the Principle of Distribution in a Socialist Market Economy

The Degree of Participation

"For the classical economists, and especially for Marx, the study of Political Economy and the analysis of exchange-value necessarily started from those socio-economic conditions that shaped the class relations of society." [Dobb, Maurice: 1972, p. 31]

"The produce of the earth -- all that is derived from its surface by the united application of labour, machinery, and capital, is divided among three classes of the community; namely, the proprietor of the land, the owner of the stock or capital necessary for its cultivation, and the labourers by whose industry it is cultivated."

"But in different stages of society, the proportions of the whole produce of the earth which will be allotted to each of these classes, under the names of rent, profit, and wages, will be essentially different ..."

To determine the laws which regulate this distribution, is the principal problem in Political Economy ..." [Ricardo, David: 1817, Preface].

Marx defines classes like Ricardo on the basis of their source of income. In his "Critique of the Gotha Programme" Marx emphasizes that "the instruments of labor are the monopoly of the capitalist class" as well as of the class of the landowners and "the resulting dependence of the working class is the cause of misery and servitude in all forms."

The classical model of distribution

For simplicity we distinguish here only between capital (K) and labour (L) as the sources of income (Y) of the class of capitalists, profit (P) and of wages (W) as the income from labour.

We can express total income of society Y as the sum of incomes by capitalists Y_C and workers Y_L as:

Y = Y_C + Y_L
which consist of the incomes of the members of each class. The individuals i = 1 to m as capitalists receive profits from their ownership of capital:


and the individuals i = m + 1 to n receive wages from their work:

This is the classical model of the distribution of personal income amongst the members of different classes of society.

The traditional socialist view is that only by nationalizing the means of production this division of society into classes can be overcome.

However, if we search for the principle of distribution for a socialist economy based on private means of production there has to be an improved cross-distribution of wealth and income to abolish the antagonism of class. To analyze such a distribution we redefine the classical model as follows:

The model of cross-distribution of income amongst classes

Individuals receive income from the different sources capital and labour. As labour is the primary source and capital yields income only because of its relative scarcity we measure the income from capital as a proportion q of income from labour.

q is in essence the same as the rate of surplus value m' which expresses the surplus s as proportion of variable capital v which has produced it.

m' = s / v.

However in our context q is the relation of profits to wages in individual income. q_i is called the degree of participation.

For total income Y = P + W we express profits as a part of wages by q = P/W and therefore total income:

Y = W (1 + q)

For individuals the cross-distribution of incomes from different source can now be expressed by their q_i.

In a capitalist society we cannot define q_i for some capitalists as there is no income from work.
For this class the only source of income is profits as in the classical model:

But for the rest of society, the individuals i = m + 1 to n who participate in work we can define their degree of participation q_i,

There exist some q_i which are almost infinite as some capitalists income from work is very small in relation to their income from profits.

For some persons q_i < 0 as they have to pay some debts from their wages.

Many persons receive only wages and their q_i = 0 which is the classical case.

It is interesting to note that we are using the same tool, the rate of surplus value m' which Marx has used to analyze exploitation. Here we apply it to the distribution of personal income from different sources to find the solution for a socialist market economy.

The distribution of personal income in a capitalist market economy can be characterized by a distribution which has approximately the following shape:

This distribution is known as the Pareto distribution [Pareto, Vilfredo, 1896]. The point P characterizes the level of income I or higher that has been obtained by N persons. The higher the income, the less persons obtain that level of income or more.

The shape of this distribution is not only determined by the unequal distribution of wealth and the profits deriving from this but also the income from work or income from earnings is highly unevenly distributed (See Lydall 1968).

Differences of personal income due to differences of ownership of the sources of income can be measured by the degree of participation q_i.

Different intensities of using wealth in order to obtain income and different intensities to work are some other factors determining the distribution. We are concerned here only with those conditions which define the classless society. Great inequality amongst persons may cause other forms of dependence. That is of importance too as expressed in demands for "equal opportunity" and "solidarity". But these issues are not considered here. We are searching the principle of distribution which overcomes the antagonisms of the classes and this can be expressed in terms of the degree of participation q which is the rate of surplus value m' applied to personal income.

The variation of q amongst persons characterizes the degree of socialisation of society.

If there exist differences in the degree of participation persons with a q_i greater q "exploit" the persons having a q_i less than average.

The antagonism amongst classes is overcome if the individual degrees of participation are equal to the average degree of participation, if the composition of income of each is equal to the composition of total income of society,

q = q_i for i = 1 ... n,

This is the fundamental principle of distribution for a socialist market economy.

Capitalism recognizes the importance of means of production to increase the productivity of labour but the ruling class exploits the working people by owing and controlling these means of production and deriving a monopoly income from their relative scarcity.

The transition to socialism consists in abolishing this monopoly of ownership and to establish a relationship between work and ownership of wealth.

It is obvious from the analysis above that a transition from a planned socialist economy to a socialist market economy is in principle much easier(although it has failed in the Soviet Union) than the transition from a capitalist system to a socialist market economy.

The objective of socialist policies is to increase the degree of socialisation. From the analysis it is obvious that this is only possible by reducing the q_i's of those persons who's relative income from wealth in relation to earnings is above the average. This is not an abandoning of the class struggle as some communists proclaim but indeed an intensification of it! It also means a direct improvement of the wealth of the members of the working classes. On the other hand conservative policies do not envisage to achive an equalization of the q_is but only a relative improvement of wealth of their membership, usualy on the cost of the working class. This policy is sometimes called striving for 'people's capitalism'.

Socialist policies to improve the degree of socialisation are focusing on the comand over productive capital, democratizing the economy and not on living from profits; it is not 'people's capitalism' but it is the policy of democratic socialism proper.

The average degree of participation that is to say the rate of surplus value of the economy is approximately q = 0.33% in OECD countries.

Bibliograhie:

Dobb, Maurice:	Theories of Value and Distribution since Adam Smith. Ideology and Economic Theory. Cambridge: Cambridge University Press, 1972.
Lydall, Harold:	The structure of earnings. Oxford: Clarendon Press, 1968.
Marx, Karl:	Critique of the Gotha Programme. London: 1875.
__, __	Das Kapital. Kritik der Politischen Ökonomie. Hamburg: Otto Meissner, 1867.
Pareto, Vilfredo:	La courbe de la répartition de la richesse. Lausanne, 1896.
Ricardo, David:	"On the Principles of Political Economy and Taxation", "The works and Correspondence of David Ricardo", Vol. I, ed. by Piero Sraffa with the collaboration of M. H. Dobb. Cambridge: Cambridge University Press, 1951.

Klaus Hagendorf, Amsterdam, 27.12.2002

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