The Reswitching Pattern: A Fluke Case For Macaulay's Duration

This post documents an allusion to a reswitching pattern in a paper by Saverio Fratini.

Fratini refuted Cachanosky and Lewin' attempt to create a justification for Austrian capital theory and, thus, for the Austrian business cycle theory. Their justification is built on Macaulay's Duration as a measure of the average period of production. Duration is the elasticity of a stream of costs with respect to the interest factor (1 + r).

Capitalists want to adopt a technique with a longer Duration around any switch point. But an increased Duration can be associated sometimes with an increased output per worker and sometimes with a decreased output per worker. Austrian capital theory remains unfounded, ad hoc, and arbitrary. So does the Austrian theory of the business cycle.

The working paper version of Fratini's refutation is not behind a paywall. Fratini considers a flow-input, point-output technology, in which two techniques are available for producing a single consumer good: Fratini writes:

Proposition 3. Let R^* be an interest factor such that p_a(R^*) – p_b(R^*) = 0. In a small neighbourhood of R^*, if dp_a/dR – dp_b/dR ≠ 0, then a rise in the rate of interest entails a change of the method in use and the incoming method has an average period of production shorter than the outgoing one.

In the above proposition:

• r is the interest rate.
• R = (1 + r) is the interest factor.
• R^* is the interest factor t a switch point.
• p_a(R) is the cost of producting the commodity with technique A when the interest factor is R.
• p_b(R) is the cost of producting the commodity with technique B when the interest factor is R.

What happens when the derivatives of the cost of producing the commodity, with respect to the interest factor, are equal at the switch point? Then the techniques do not switch at the switch point. Two wage curves are tangent at a switch point. I call this fluke case a 'reswitching pattern'.

I Did Not Invent The Concept Of Fluke Switch Points

I find myself writing about generic switch points, fluke switch points, anomalous switch points, fake switch points, normal switch points, and 'perverse' switch points. I do not seem to have definitions with the precision of those in mathematical analysis.

A switch point is a fluke when any perturbation of some parameters, such as coefficients of production, destroys defining features of the switch point. The concept of a fluke goes back to the 1966 symposium on capital theory in the Quarterly Journal of Economics:

"If, by a fluke more than one switch of technique happened to take place at exactly the same point, the nonzero columns [of the matrix formed by the difference of two Leontief matrices] would be more than one" (Pasinetti 1966: 511).

Other participants recognize this fluke case in which four wage curves intersect at a switch point, with processes replacing one another in two industries:

"'Adjacent" techniques on two sides of a switching point of a switching point will usually differ from each other only with respect to one activity" (Bruno, Burmeister, Sheshinski 1966: 542).

Two wages curves tangent at a switch point is another fluke case. A perturbation leads to either the reswitching of techniques or of one cost-minimizing technique around the rate of profits at which the switch point formerly existed. Other fluke cases arise when a switch point exists at the maximum wage or the maximum rate of profits:

"Cases with multiple roots or cases in which the curves cross only at end points... These ... are cases which one technique can be ignored since it is dominated" (Bruno, Burmeister, Sheshinski 1966: 534).

Pierangelo Garegnani recognizes the possibility of the fluke case with two wage curves tangent:

"The possibility that, at r^* and r^**, the two wage curves touch without intersecting is excluded...” (Garegnani 1966: 567).

Fluke switch points exist in both models of single and joint production. Vienneau (2021) examines fluke switch points in pure fixed capital models, while Vienneau (2022) partitions a parameter space, with fluke switch points, in a model of extensive rent. Vienneau (2024) looks at fluke switch points in a model of non-competitive markets with single production. The characterization of fluke switch points is useful for analyzing structural economic dynamics with a choice of technique.

References

Bruno, Michael, Edwin Burmeister, and Eytan Sheshinski. 1966. The nature and implications of the reswitching of techniques. Quarterly Journal of Economics 80(4): 526-553.

Garegnani, P. 1966. Switching of techniques. Quarterly Journal of Economics 80(4): 554-567

Pasinetti, Luigi L. 1966. Changes in the rate of profit and switches of technique. Quarterly Journal of Economics 80(4): 503-517.

Vienneau, Robert L. 2021. Fluke switch points in pure fixed capital systems. Centro Sraffa working papers n. 48.

Vienneau, Robert L. 2022. Reswitching in a model of extensive rent. Bulletin of Political Economy 16(2): 133-146.

Vienneau, Robert L. 2024. Characteristics of labor markets varying with perturbations of relative markups. Review of Political Economy 36(2): 827-843.