Anomalous Switch Points

This post is to remind me that I have discovered some anomalous switch points. I am introducing the concept of an anomalous switch point here.

Consider a switch point in a model of single production, with inputs of labor and circulating capital alone. At a (generic) switch point, one process replaces another in an industry that produces a commodity that exists in both techniques. Other processes can be introduced or removed if some capital goods are used only in one of the two processes for the common industry. The switch point is on the wage frontier formed by the outer envelope of the wage curves for all techniques. Two wage curves intersect at the switch point.

An anomalous switch point differs in some property from a generic switch point in models of circulating capital alone.

The anomalous switch points under consideration here are not flukes. A fluke switch point is one in which any permutation of some parameters destroys the qualitative property under consideration for the switch point. A switch point in which two wage curves are tangent on the frontier is an example of a fluke. A switch point in which two processes are replaced, one in each of two industries that exist in the techniques with intersecting wage curves. In this example with both types of flukes, and more, four wage curves intersect at switch point for the second kind of fluke.

Anomalous switch points include:

• A switch point in which the same processes are operated for both techniques: This example is one of extensive rent. The levels of operation of the processes and the land that is scarce vary around the switch point.
• Another switch point in which the same processes are operated for both techniques: This example is one of intensie and extensive rent with multiple agricultural commodities.
• A switch point along a single wage curve: This example combines fixed capital and extensive rent. Two techniques, Nu and Omicron, have the same 'solving system' and, hence, the same wage curve. The techniques differ in the economic life of machine when operated on scarce land that pays extensive rent.

Other switch points can be considered anomalous. D'Agata (1983) has examples with non-unique and non-exisitng cost-minimizing techniques in a model of intensive rent. Two techniques are cost-minimizing before a switch point. No cost-minimizing technique exists after the switch point. I think D'Agata's examples are more challenging to Sraffa's price theory than mine. Woods (1990) also has an example in joint production, without rent.

A non-fluke switch point in a model of joint production in whcih three wage curves intersect is another anomalous switch point. Bidard and Klimovsky have a genuine switch point like this in the paper in which they introduce the concept of a fake switch point. I think I also have examples in my models that combine intensive and extensive rent.

I do not consider anomalous a switch point on the wage frontier, in which the frontier is not the outer envelope of wage curves. The wage frontier must be the outer envelope in models of single production, but researchers in joint production established last century that this property need not hold in models of joint production, including in models of extensive rent.

Fake switch points can also be anomalous. With joint production, a question arises about which process should be replaced when a new process is introduced. A fake switch point, in a model of joint production, is an intersection of (non-tangent) wage curves in which the cost-minimizing technique does not change. The intersection's location and existence depend on the numeraire. The price of a commodity produced under both techniques whose wage curves intersect varies among techniques.

A fake switch point can be anomalous in that it deviates from properties of Biard and Klimovsky's example:

• Example fixed capital and extensive rent: Prices of commodities produced under both techniques do not vary among techniques at switch points. Price and rent vary between techniques only for non-commodities (that are free) under the non-cost-minimizing technique.

Technical terms: Switch point, Anomalous switch point, Fake switch point, Anomalous fake switch point, Fluke switch point.

References

• D’Agata, A. 1983. The existence and unicity of cost-minimizing systems in intensive rent theory, Metroeconomica 35: 147-158.
• Bidard, Christian and Edith Klimovsky. 2004. Switches and fake switches in methods of production. Cambridge Journal of Economics 28 (1): 88-97.
• Vienneau, Robert L. 2024. Characteristics of labor markets varying with perturbations of relative markups. Review of Political Economy 36(2): 827-843.
• J. E. Woods (1990) The Production of Commodities: An Introduction to Sraffa, Humanities Press International