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'.