Linear Classical Models of Production

Central elements of classical analysis are the concept of production as a circular flow and the related concept of surplus product left after the wage goods and what is nec­essary for the replacement of the used-up means of production have been deducted from the annual output.

Robert Torrens in his Essay on the External Corn Trade (1815) clarified that the concept of surplus provides the key to an explanation of the rate of profits. Growth in the model by Torrens is both linear and endogenous; the rate of growth depends on the general rate of profits and the propensity to accumulate. The same can be said of Marx’s theory of expanded reproduction in chapter 21 of volume II of Capital (Marx 1885 [1956]). In it Marx studied the conditions under which the system is capable of reproduc­ing itself on an upward spiralling level: the economy expands at an endogenously deter­mined rate of growth. This rate depends on the proportion of the surplus value ploughed back into the productive system to increase the scale of operation. Marx stressed that the accumulation of capital is “an element immanent in the capitalist process of production” (ibid.: 497, emphasis added). For, “the aim and compelling motive of capitalist produc­tion” is “the snatching of surplus-value and its capitalisation, i.e., accumulation” (ibid.: 507). In Marx’s analysis, this theory is only a first logical step toward a proper theory of accumulation. Here we cannot deal with the latter and Marx’s “law” of a falling ten­dency of the rate of profits in volume III of Capital.

The Russian mathematician Georg von Charasoff elaborated on Marx’s analysis and was possibly the first to provide a clear statement of the fundamental duality relationship between the system of prices and the rate of profits on the one hand, and the system of quantities and the rate of growth on the other (see Charasoff 1910). He developed his main argument within the framework of an interdependent model of (single) production exhibiting all the properties of the later input-output model, and which is fully speci­fied in terms of use values (rather than labour values as in the case of Marx) and labour needed per unit of output (see Kurz and Salvadori 1995: ch. 13).

John von Neumann (1945) in a paper first published in German in 1937 and then translated into English in 1945 elaborated the by far most sophisticated linear model of endogenous growth. In it von Neumann assumed that there are n goods produced by m constant returns-to-scale production processes. There is a problem of the choice of technique, which consists of establishing which processes will actually be used and which not, being “unprofitable”. Von Neumann (1937 [1945]: 1-2) took the real wage rate, consisting of the “necessities of life”, to be given and paid at the beginning of the uniform period of production, that is, he considered wages as a part of the capital advanced and thus as a part of the physical real costs of production. In addition, he assumed “that all income in excess of necessities of life will be reinvested”. In von Neumann’s model the rate of growth is determined endogenously. He set aside the problem of scarcity of all non-accumulable factors of production: while all primary factors other than labour (that is, all natural resources) were taken to be available at whichever amount was needed at zero price, labour was assumed to be available at the required amount at a given real wage rate.