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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:
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| 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
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| Figure 2: Partitions of the Parameter Space Sixth Double-Fluke Switch Point |
| 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.
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| Figure 3: Rate of Profits and the Wage at Certain Fluke Switch Points |