Sraffa and von Neumann

The interpretation of the classics presented here first appears in Sraffa’s 1951 introduc­tion to Ricardo’s Principles (1951-73: I); it subverts the Marshallian view of a continuity between Ricardo and marginalist theory, but its implications take time to be appreciated; important contributions to this effect are by Garegnani (1960, 1987, 2007).

To summarize an immense “Sraffian” formal literature up to the 1990s (among the many contributors one should mention at least Pasinetti, Garegnani, Schefold, Steedman, H.D. Kurz, Salvadori, Parrinello, Bidard), the consistency of the surplus approach’s determination of r on the basis of data (1), (2) and (3) listed in the section on Adam Smith is fully confirmed for the cases of simple production, fixed capital (treated as a special case of joint production as suggested in the past by Torrens), and extensive dif­ferential land rent, without or with choice of techniques. Given the production methods among which firms can choose, the “effectual demands”, the quantities and qualities of land, and either the real wage or the rate of profits, there is only one economically sig­nificant solution to the equations determining adopted methods, normal prices (prices of production), and the other distributive variables, such that no non-adopted method allows extraprofits; “truncation” of fixed capital (that is, when to discard durable capital goods) is also determined.

A result by Sraffa of particular interest to the historian of economic theory is that it is possible to determine the rate of profits as a material ratio, even when physical homo­geneity between input and output is not obtained in any sector, by assuming a notional change of industry dimensions such that the economy produces a composite commod­ity with the composition required to render the rate of physical surplus (if wages are advanced and included in the production coefficients) or the rate of net product (if wages are paid in arrears) uniform for all basic commodities. When producing this Standard Commodity, product and capital are amounts of the same composite commodity, as in Ricardo’s “corn model”. The Standard Commodity, essential to Sraffa for several dem­onstrations because the Perron-Frobenius theorem was unknown to the mathematicians he consulted in Cambridge, renders explicit a mathematical property of the coefficients matrix, its dominant eigenvalue with the associated right and left eigenvectors, which determines the maximum rate of profits and of growth with the associated relative prices and relative industry proportions, as already noted in the past by long forgotten or mis­understood authors such as Charasoff, Remak or von Neumann (see Kurz and Salvadori 1995: ch.

Of these the most important is von Neumann, who in a paper presented at a seminar at Princeton University in 1932 introduced inequalities and mathematical tools from game theory into a linear model of production and pricing (with given wages advanced and included in the technical coefficients), in order to admit joint production and choice among a number of linear alternative production methods. Von Neumann assumes the “rule of free goods” (price zero of commodities in excess supply, a possibility with joint production), determines the demand side by assuming that all profits are reinvested and that the growth rate is maximized, and proves that the rate of profits equals the growth rate. His paper, published in 1937 in German, was translated into English in 1945 with the greatly altered title “A model of general economic equilibrium” which helped its misinterpretation as a neoclassical general equilibrium model motivated by an interest in optimal growth; accordingly it gave rise to the studies culminating in “turnpike theorems”. After the rediscovery, with Sraffa and Garegnani, of the structure of the classical approach, it became clear that von Neumann’s paper was internal to the classical approach (as shown by the given wages included in the technical coefficients, by the absence of any substitution in consumer demand, and by the absence of any assumption of full labour employment), probably owing to an indirect influence of Bortkiewicz. The real purpose of the paper appears to have been to determine the rate of profits as a material ratio on the basis of a given real wage, while admitting technical choice and joint production: the very last sentence of the paper determines the rate of profits as equal to the previously deter­mined maximum rate of growth; and, neglecting the commodities with zero price, the economy is producing a Standard Commodity.