I have been using fluke switch points to partition parameter spaces into regions. In each region, the analysis of the choice of technique does not qualitatively vary.
Fluke switch points have been discussed in the literature on the analysis of the choice of technique. Mostly, these mentions dismissal fluke cases, on the correct grounds that they only occur at an accidental point in the parameter space.
Here is a statement from one contribution to the famous QJE symposium:
"Cases with multiple roots or cases in which the curves cross only at end points … can be classified as irrelevant since the F[actor] P[rice] F[rontier] (envelope) is unchanged by their exclusion." -- Bruno, Michael, Edwin Burmeister, and Eytan Sheshinski. 1966. The nature and implications of the reswitching of techniques. Quarterly Journal of Economics, 80 (4): 534.
And another statement from the same contribution:
"'Adjacent' techniques on two sides of a switching point will usually differ from each other only with respect to one activity." -- Bruno, Michael, Edwin Burmeister, and Eytan Sheshinski. 1966. The nature and implications of the reswitching of techniques. Quarterly Journal of Economics, 80 (4): 542.
Here is from Garegnani's contribution:
"The possibility that, at r* and r**, the two curves touch without intersecting is excluded…" -- Garegnani, Pierangelo. 1966. Switching of techniques. Quarterly Journal of Economics, 80 (4): 567
I suppose I can look for more.
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