Monday, February 03, 2025

An Expanded Parameter Space For The Reverse Substitution Of Labor

Figure 1: A Larger Parameter Space

This post is an expansion on the first example here. It presents shortly a more comprehensive analysis of the variation in the choice of technique in the example of circulating capital in Section 2. Local perturbations of two coefficients of production are examined there. Figure 1 partitions a larger part of the space defined by these two coefficients of production. Table 1 exhibits how the cost-minimizing technique varies with the rate of profits in each region.

Table 3: Ranges of the Rate of Profits by Region
RegionRangeTechniqueNotes
10 ≤ rr1BetaReverse substitution of labor at switch point.
r1rrα,maxAlpha
20 ≤ rr1BetaSwitch point is 'non-perverse'.
r1rrα,maxAlpha
30 ≤ rrβ,maxBetaNo switch point.
40 ≤ rrβ,maxBetaNo switch point.
50 ≤ rr1AlphaSwitch point is 'non-perverse'.
r1rrβ,maxBeta
60 ≤ rr1AlphaReswitching. Second switch point exhibits capital-reverseing and the reverse substitution of labor.
r1rr2Beta
r1rrα,maxAlpha
70 ≤ rrα,maxAlphaNo switch point.

Section 2 in the previous focuses on regions 1, 2, 3, and 4. Region 3 and 4 differ in that in region 4, the wage curves intersect at a negative rate of profits greater than -100 percent. This post presents an analysis, in a model of fixed capital, much like the fluke switch point associated with the intersections of the partitions between regions 1, 6, and 7. Does checking how the variation of the analysis of the cost-minimizing technique among regions, summarized in Table 1, relates to the fluke cases defining the partitions in Figure 1 clarify that variation? Perturbations of coefficients of production, in this example of circulating capital, illustrate how reswitching can emerge, as well as the emergence of the reverse substitution of labor.

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