Robert Murphy On Sraffa: In Error

Some discussion with Peter Boettke has inspired me to point out some technical mistakes in Robert Murphy's on-line comments on Sraffa and reswitching.

I begin with Murphy's comments on reswitching. He looks at Samuelson's example in Samuelson's "Summing Up" article. Murphy implicitly suggests that reswitching is only possible in models in which a finite number of techniques are available:

"What Samuelson has done is simply invent a fictitious world in which there are only two ways of producing a particular good... Böhm-Bawerk felt that [his] story was accurate, because at any given time there are more technically efficient but very time-consuming processes 'on the shelf' that are unprofitable at the market rate of interest, but would become profitable at lower rates."

But reswitching is possible when a continuum of techniques lie along the so-called factor price frontier. That is, the possibility of reswitching is consistent with the existence of an uncountably infinite number of techiques. It is also consistent, of course, with the existence of only a countably infinite number and only a finite number of techniques.

Murphy also writes an equally informed comment on Sraffa's book, The Production of Commodities by Means of Commodities. I will adopt Austrian - in fact, Misian - terminology. Sraffa compares prices in Evenly Rotating Economies (EREs) in which the same commodities are produced with the same inputs. Under Sraffa's assumptions in the first part of his book, the construction of the so-called factor price frontier is perfectly valid mathematically. Murphy notes that Sraffa does not model utility-maximization and states that if utility maximization is introduced into the model, the location on the frontier becomes determined uniquely:

"Sraffa's techniques leave no room for the individual members of society to influence the methods of production that end up being used (whether or not there is a surplus), ultimately because there are no individuals in Sraffa's models... However, if we also require that the market rate of interest reflects the subjective premium placed by consumers on present versus future consumption—a feature lacking in Sraffa's aggregate models—then this will eliminate the multiplicity of equilibrium rates of interest."

But Murphy is, again, mathematically incorrect. Multiple equilibrium rates of interest can arise in an ERE model with utility maximization, including intertemporally.

One might look outside a model of an ERE. Murphy suggests he wants to consider models of an approach to an ERE:

"Sraffa's method of determining equilibrium prices in a surplus economy already assumes that the system has settled down at the optimum level of production in all possible lines."

The Arrow-Debreu model of intertemporal equilibrium, despite all its problems, is sufficient for my point here. In such a model of an economy not in an ERE, the equilibrium rate of interest at any point in time for loans of a given length is also not necessarily unique. Not only can multiple equilbrium rates of interest arise, so can a continuum of equilibrium interest rates, if the technology is modeled as discrete.

Why might Murphy be inclined to insist on mathematical error? Consider his statements:

"Sraffa derives results that depict a tradeoff between the real wage and rate of profits. In particular, Sraffa's analysis suggests that in a developed economy, the proportion of the 'surplus' that goes to the workers versus the capitalists is arbitrary, and not at all 'determined' by technological or economic facts... Although he was wrong to condemn interest as an unnecessary and exploitive institution, Sraffa was perfectly correct to criticize the conventional, mainstream justification of the capitalists' income."

But none of these claims, including about exploitation, are made in Sraffa's book.