Thursday, July 04, 2019

On Milana's Purported Solution To The Reswitching Paradox

Carlo Milana has posted a paper on arXiv. I was prepared to accept this paper's claims. Economists have developed price theory. Referring to Sraffian "paradoxes" and "perverse" switch points is a matter of speaking. There does not exist separate Sraffian and neoclassical versions of price theory. For a result to be "perverse", it need only contradict outdated neoclassical intuition. But it is as much a part of the mathematical economics as any other result. (It is another matter that much teaching in microeconomics is inconsistent with the mathematics.)

In equilibrium, the price of the services of each capital good in use is equal to the value of the marginal product of that good, with all prices discounted to the same moment in time. This discounting implies that the interest rate appears in a formal statement of these equations. These equalities are very different from the claim that the interest rate equals the marginal product of (financial) capital. In limited cases, one can prove something like the aggregate equality. No such thing as a marginal productivity theory of distribution, however, is restored. Milana might cite Hahn (1982) on this background.

But Milana goes further. He claims that reswitching is impossible or, at least, examples up to now are erroneous.

I think Milana's basic mistake is exposed in Salvadori and Steedman (1988). (I go through one of their examples here.) The Samuelson-Garegnani model is not a model of two-(produced) goods. The model contains as many capital goods as there are techniques. Potentially, there can be a continuum of capital goods in the model. As such, it is meaningless to require the price of the capital goods used in each technique that is cost-minimizing at a switch point to be equal to one another. That is analogous to requiring the price of a ton of iron be equal to the price of a ton of tin.

For some reason, Milana does not discuss examples of reswitching in flow-input, point output models, such as in Samuelson (1966). Nor does he acknowledge, as I read him, valid examples in which the same n commodities are produced in all techniques, and all commodities are basic in all techniques. (Does he say anything at all about the distinction between basic and non-basic commodities?) At a non-fluke or generic switch point, in such a framework, the two techniques that are cost-minimizing differ in a process in exactly one industry.

Milana should read and reference Bharadwaj (1970), as well as Bidard and Klimovsky (2004) on fake switches in models of joint production. Other Linear Programming formulations are available for considering the choice of technique. Vienneau (2005) presents one. What does Milana have to say about the direct method for analyzing the choice of technique in Kurz and Salvadori (1995)? I briefly provide a survey of different analysis in Vienneau (2017), as well as an algorithm for finding the cost-minimizing technique? Are all these approaches in error?

  • Khrishna Bharadwaj. 1970. On the maximum number of switches between two production systems. Schweizerische Zeitschrift fur Volkswortschaft and Statistik (4): 401-428. Reprinted in Bharadwaj 1989. Themes in Value and Distribution: Classical Theory Reappraised, Unwin-Hyman.
  • Christian Bidard and Edith Klimovsky. 2004. Switches and fake switches in methods of production. Cambridge Journal of Economics 28:89-97
  • Frank Hahn. 1982. The neo-Ricardians Cambridge Journal of Economics 6:353-374.
  • H. D. Kurz and N. Salvadori. 1995. Theory of Production: A Long-Period Analysis. Cambridge University Press.
  • Carlo Milano. 2019. Solving the Reswitching Paradox in the Sraffian Theory of Capital. arXiv:1907.01189
  • Neri Salvadori and Ian Steedman. 1988. No reswitching? No switching! Cambridge Journal of Economics 12: 481-486.
  • Paul A. Samuelson. 1966. A Summing Up. The Quarterly Journal of Economics 80 (4): 568–583.
  • Robert Vienneau. 2005. On labour demand and equilibria of the firm. The Manchester School 73(5): 612-619.
  • Robert Vienneau. 2017. The choice of technique with multiple and complex interest rates. Review of Political Economy 29(3): 440-453.


monerty said...

Hello prof.vienneau recently an argument was made that if you take into account not a single interest rate but a term structure of interest rates reswitching disappears its in this paper in page 242 its by another austrian, it also later repeats osbornes responce which you have already adressed , so i was wondering whats your view on this responce ?

Carlo Milana, Birkbeck College, University of London said...

Robert Vienneau’s comment is out of track. It is the usual game of attributing false statements to the counterparty. The false statement attributed to my paper is that, at a switch point between two techniques of production, “the price of a ton of iron be equal to the price of a ton of tin”! By contrast, I recalled Sraffa’s (1960, p. 98) statement that

“two different methods of producing the same basic commodity can only co-exist at the points of intersection (that is to say, at those rates of profits at which the prices of production by the two methods are equal)."

What I instead stated by quoting Sraffa is completely different from Vienneau’s reading of my words, that is, at a switch point between two techniques, the price of a ton of iron is the same with both techniques, and the price of a ton of tin is the same with both techniques.

Unfortunately, the comment goes further. It wonders how so many authors are found to be mistaken, and it wonders why Samuelson’s (1966) has not been mentioned in my paper. The reason is simpler than what is implied. Space limitations of my paper led me to discuss only intersectoral models such as those considered throughout Sraffa’s (1960) book, except his chapter 6, where an intertemporal model is considered. The intertemporal model is indeed discussed in my other paper entitled “Refuting Samuelson’s (1966) capitulation in the Cambridge controversy on capital theory”.

Vienneau’s comment ends by mentioning several direct methods proposed in the literature for analysing the choice of techniques and asks: “are all these approaches in error?" Let me summarize my conclusion on the Sraffian approaches. All these approaches fail to recognize that the return of techniques is the consequences of the return of the ranking in relative input prices over the feasible range of real wage (or interest rate). Sraffa and almost all his followers failed to see the key concept of “rental prices of capital goods”. This is surprising since the rental prices of capital goods are, nevertheless, always present in their models as well as in the neoclassical models of cost and production functions.

The rental prices are non-linearly related to the interest rate (or profit rate) so that they may return to previous levels as the latter changes. It is, in fact, the rental price of a capital good relative to labour wage that is equal to the ratio of respective (negative) marginal factor productivities, rather than rate of interest relative to labour wage. And the Sraffian reduced form of the real wage-interest frontier is a counterfeit imitation of the correct concept of real factor-price frontier (and even Samuelson, 1966 has been driven into this oversight).

The return of rental prices relative to labour wage over the range of the interest rate is the movement that explains the apparent phenomenon of reswitching of cost-minimizing techniques over the range of interest rate or real labour wage. But this reswitching phenomenon disappears in the coordinate space of rentals relative to wage, thus confirming the validity of the pure marginalist theories of factor rewards, in full contradiction of the Sraffians’ oversight.