Recap For A Triple -Switching Example

Figure 1: Actual and Stylized Partitions of Parameter Space with Triple-Switching

This post is a continuation of this series of posts.

The partitioning of the parameter space by fluke switch points in these posts can be combined into one picture. The left pane in Figure 1 illustrates. The dashed line is a ray from the origin, discussed below. I find this complete picture for this example hard to perceive by eye. The right pane provides a highly stylized presentation of the partitions, rotated and stretched. The partitions are not straight lines on the left. The boundary between regions 1 and 5 is tangent to the boundary between regions 1 and 2 at the point of intersection. The boundary between regions 3 and 4 is likewise tangent to the boundary between regions 2 and 4 where they intersect. The two boundaries between regions 6 and 7 become tangent at their point of intersection. Region 6 is for triple-switching. It adjoins regions 3, 5, and 7. Regions 3 and 5 are examples of reswitching. Their borders with region 6 have fluke switch points on the axis for the rate of profits and on the wage axis, respectively. The borders between regions 6 and 7 are associated with fluke switch points in which two wage curves are tangent. Reswitching in regions 3 and 5 appears in a fairly generic fashion. Reswitching can also appear from perturbations of coefficients of production, where the region in parameter space corresponding to reswitching is not adjacent to a region in which triple-switching occurs.

Table 1: Cost-Minimizing Technique byRegion

Region	Cost-Minimizing Technique	Notes
1	Alpha	No switch point.
2	Alpha, Gamma	Around the switch point, a lower rate of profits is associated with a LESS round-about technique and greater output per worker.
3	Gamma, Alpha, Gamma	Around the second switch point, a lower rate of profits is associated with a LESS round-about technique and LOWER output per worker.
4	Gamma	No switch point.
5	Alpha, Gamma, Alpha	Around the first switch point, a lower rate of profits is associated with a LESS round-about technique. Around the second switch point, a lower rate of profits is associated with LOWER output per worker.
6	Gamma, Alpha, Gamma, Alpha	Around the second switch point, a lower rate of profits is associated with a LESS round-about technique and LOWER output per worker.
7	Gamma, Alpha	Around the switch point, a lower rate of profits is associated with a more round-about technique and greater output per worker.

The example demonstrates that an increase in the roundaboutness of the cost-minimizing technique is independent of its capital-intensity. For the traditional marginalist story, a lower rate of profits around a switch point is associated with a choice of technique with greater capital-intensity and greater output per worker. For the traditional story from the Austrian school, a lower rate of profits around a switch point is associated with the adoption of a more roundabout technique.

Yet all four entries in the grid in Table 2 are populated by switch points in the example. Consider the switch point in region 2 or the first switch point in region 5. Around these switch points, a lower rate of profits is associated with the adoption of a more capital-intensive, but a LESS roundabout technique. The less roundabout and more capital-intensive technique has a greater output per worker. These switch points populate the lower left entry in the table. The second switch point in region 5 populates the upper right entry. Around this switch point, a lower rate of profits is associated with the adoption of a more roundabout technique with LOWER output per worker. The first switch point in region 3 and the switch point in region 7 populate the upper left entry. They happen to be consistent with these old theories. The second switch point in region 3 fills the entry in the lower right in the table. Roundaboutness and capital-intensity move together, but against the intuition of outdated marginalist and Austrian school economists. A lower rate of profits is associated with a LESS round-about technique and LOWER output per worker. The first and third switch points in region 6, in which triple-switching occurs, are like the switch point in region 7. The second switch point is like the second switch point in region 3. Roundaboutness and capital-intensity do not seem to have much to do with one another.

Table 2: Lower Rate of Profits around a Switch Point

	Traditional Marginalist Story	'Perverse' Marginalist Story
Traditional Austrian Story	Greater net output per worker	Smaller net output per worker
	More roundabout technique	More roundabout technique
'Perverse' Austrian Story	Greater net output per worker	Smaller net output per worker
	Less roundabout technique	Less roundabout technique

Figure 2: Structural Dynamics for an Example of Triple-Switching

These posts explore structures in parameter spaces that might not be immediately visible in empirical regularities at surface levels. Since distribution is not specified, the model of the choice of technique is open. The impact on the dynamics of market prices of coefficients of production supporting triple-switching is not clear. Such temporal dynamics, one might expect, depend on the speed with which capitalists adopt processes adapted to new technology and distribution, as compared to the speed with which technology improves. Market dynamics might depend on the history of such adjustments, as reflected in fixed capital remaining from previous adjustments. The size of the extra profits obtainable by these adjustments is another consideration. Even if triple-switching were quickly manifested in struggles over the distribution of income and in market dynamics, the partitioning of parameter spaces by fluke switch points suggests that triple-switching might be rare. It only occurs in specific examples of structural dynamics.

These posts demonstrate that triple-switching can arise through innovations in technology. The illustrated traversal of the parameter space is not the only way. Reswitching can arise as here and as adjacent to a triple-switching example. Likewise, triple-switching can arise adjacent to an instance of quadruple-switching. One can see this by considering generalizations of Figure 2. Each instance with more switch points is less likely to correspond to a region in the parameter space formed by coefficients of production or related parameters. At any rate, the number of partitions in parameter space increases, and their configurations are more complicated. In the example, triple-switching arises from technological innovation. But further innovation in the same direction removes the possibility of triple-switching. This result applies to reswitching, and generalizes to quadruple-switching, and so on. Regions with multiple switch points are transient, arising as one technique replaces another as dominant, whatever the distribution of income.

The example examined in the main text has also demonstrated that the degree of roundaboutness is independent of the capital-intensity of a technique. Keynes had a point:

"It is true that some lengthy or roundabout processes are physically efficient. But so are some short processes... Moreover there are all sorts of reasons why various kinds of services and facilities are scarce and therefore expensive relatively to the quantity of labour involved. For example, smelly processes command a higher reward, because people will not undertake them otherwise. So do risky processes. But we do not devise a productivity theory of smelly or risky processes as such." -- Keynes (1936)

This series of posts re-iterates that the rate of profits is not an index for the relative scarcity of capital. A lower rate of profits need not be associated with a technique that is either more capital-intensive or more roundabout. Likewise, the wage is not an index for the relative scarcity of labor.

Previous research suggests that perturbations in relative markups can also bring about the same variations in the analysis of the choice of technique as those that result from perturbations in coefficients of production (Vienneau 2024a). Hence, triple-switching also seems to be possible as a result of long-lasting variations in relative markups.