A Sixth Double-Fluke Switch Point For A Triple-Switching Example

Figure 1: Extra Profits at Gamma Prices for the Sixth Double-Fluke Switch Point

This post is a continuation of this series of posts.

In the last double-fluke case, the three switch points between Alpha and Gamma coincide as a ingle switch point. Figure 1 illustrates, while Figure 2 depicts how the parameter space is partitioned around this double-fluke case. Region 7, in which one switch point occurs, is connected. At the point corresponding to the double-fluke case, the two boundaries between regions 6 and 7 are tangent. Schefold's example is at a point, (φ t, σ t)=(1,1⁄2), in the thin wedge for region 6 in Figure 2. I did not find that points in the parts of region 6 in previous posts had more visually compelling wage frontiers than the point that Schefold found

Figure 2: Partitions of the Parameter Space Sixth Double-Fluke Switch Point

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 partitions of parameter space show that two values of σ t can be found as functions of φ t, where the corresponding wage curves are tangent at a switch point. Figure 3 plots the rate of profits and the wage for the switch points for these combinations of parameters. One set of three switch points is shown as a solid line and the other as a dashed line. The non-repeating switch point, for each set, is not a fluke except when on an axis or at the extreme right. The switch points for each set of parameters converges to a single switch point, with an increasing φ t. The convergence is complete at the double-fluke case.

Figure 3: Rate of Profits and the Wage at Certain Fluke Switch Points