A Non-Reswitching Theorem Inapplicable To Non-Competitive Markets?

Consider a circulating capital model of the production of commodities. A non-reswitching theorem exists:

Theorem: Suppose a commodity exists which is a basic commodity in all techniques. And a smooth, continuously differentiable production function exists for producing that commodity. Then the reswitching of techniques cannot arise.

Marglin (1984: 285-286) states a theorem like this in which continuously differentiable production functions exist for all commodities. He also states:

"Once again, a result I thought to be original turned out not to have been. Reviewing the literature for these notes, I found a proof of the impossibility of reswitching in a continuous-substitution framework in Burmeister and Dobell (1970)." – Marglin (1984: 542).

Marglin's proof is one by contradiction. Capital-reversing is still possible under the assumption of these theorems.

Pasinetti and Scazzieri (2008) find the theorem in Bruno, Burmeister, and Sheshinski (1966), which I do not recall. They, in turn, attribute the theorem to Martin Weitzman and Robert Solow. Pasinetti and Scazzieri doubt the validity of the theorem.

"It is worth noting that Weitzman–Solow's theorem is simply a consequence of the idea that, in the case of a commodity produced by a neoclassical production function, each set of input–output coefficients ought to be associated in equilibrium with a one-to-one correspondence between marginal productivity ratios and input price ratios. No ratio between marginal productivities would be associated with more than one set of input prices, and this is taken to exclude the possibility that the same technique be chosen at alternative rates of interest, and thus at different price systems. The Weitzman–Solow theorem is at the origin of a line of arguments that has been followed up by a number of other authors, such as David Starrett (1969) and Joseph Stiglitz (1973). These authors have pursued the idea that 'enough' substitutability, by ensuring the smoothness of the production function, is sufficient to exclude reswitching of technique. However, non-reswitching theorems of this type involve that, for each technique of production, the capital stock may be measured either in physical terms or at given prices. For in a model with heterogeneous capital goods, if we allow prices to vary when the rate of interest or the unit wage are changed, there is no reason why the same physical set of input–output coefficients might not be associated with different price systems: even in the case of a continuously differentiable production function, the marginal product of 'social' capital cannot be a purely real magnitude independent of prices. Once it is admitted that 'in general marginal products are in terms of net value at constant prices, and hence may well depend upon what those prices happen to be' (Bliss, 1975, p. 195), it is natural to allow for different marginal productivities of the same capital stock at different price systems. It would thus appear that reswitching of technique does not carry with it any logical contradiction even in the case of a smoothly differentiable production function." Pasinetti and Scazzieri (2008)

I do not know about that. But I have never been clear on how substitutability is supposed to justify marginalist theory.

Suppose rates of profits differ among industries. And that the ratios of rates of profits among industries are stable in the long run. I have shown that the arguments of the Cambridge capital controversy extend to such non-competitive markets. In my paper, I had a reswitching example, in a discrete technology, that did not arise in competitive markets.

I conjecture that the non-reswitching theorem for continuous-substitution does not apply to non-competitive markets.

References

• Bruno, M., Burmeister, E. and Sheshinski, E. 1966. The nature and implications of the reswitching of techniques. Quarterly Journal of Economics 80, 526–53.
• Burmeister, Edwin and A. Rodney Dobell. 1970. Mathematical Theories of Economic Growth. New York: Macmillan.
• Marglin, S. A. 1984. Growth, Distribution, and Prices. Harvard University Press.
• Pasinetti, L. L. and Roberto Scazzieri, R. 2008. Capital theory (paradoxes). The New Palgrave.
• Starrett, D. 1969. Switching and reswitching in a general production model. Quarterly Journal of Economics 83, 673–87.
• Stiglitz, J. 1973. The badly behaved economy with the well-behaved production function. In Models of Economic Growth, ed. J.A. Mirrlees and N.H. Stern. London: Macmillan.