Variations In An Analysis Of Intensive Rent With One Type Of Land (Part 2/2)

5.0 Fluke Cases

This post is a continuation of this one. This is a numeric example of intensive rent. Here I present five fluke cases before depicting how the analysis of the choice of technique varies with the full range of relative markups in agriculture.

5.1 Switch Point at Maximum Scale Factor for Epsilon

In the first fluke case, the wage curves for Alpha and Delta intersect at the maximum scale factor for the rate of profits for Delta (Figure 7). Figure 8 displays the graphs of the rent curves. At any larger scale factor, rent in Delta would be negative. This fluke case is associated with a qualitative change in the range of the scale factor for the rate of profits in which no cost- minimizing technique exists. The wage frontier consists of the wage curves for the Delta and Epsilon techniques up to the switch point between them. The wage frontier ends there. No technique is cost-minimizing for a scale factor between this switch point and the maximum scale factor for the rate of profits for Alpha.

Figure 7: Wage Curves for First Fluke Case

Figure 8: Rent Curves for First Fluke Case

This fluke case is associated with the disappearance of a range of the scale factor, for smaller relative markups in agriculture, in which only Alpha and Delta have positive scale factors for the rate of profits, and Delta has a positive rent. For a larger relative markup, a range of the scale factor appears in which only Alpha and Epsilon have positive scale factors, and Epsilon has a positive rent. A cost-minimizing technique exists in neither range.

5.2 Alpha vs. Epsilon Switch Point at Zero Wage

Another fluke case exists when the wage curves for Alpha and Epsilon intersect at a wage of zero. Figure 9 shows the wage curves, and Figure 10 shows the rent curves. In the last range for the scale factor, only Alpha can be under consideration for the cost-minimizing technique. For a smaller relative markup in agriculture, Epsilon is not eligible in this range because it would have a negative scale factor. For a larger relative markup, Epsilon is not eligible because it would have a negative rent.

Figure 9: Wage Curves for Second Fluke Case

Figure 10: Rent Curves for Second Fluke Case

These two fluke cases change some characteristics of the range of the scale factor of the rate of profits in which no cost- minimizing technique exists.

5.3 Switch Point for Three Techniques

For the next fluke case, all three wage curves, for Alpha, Delta, and Epsilon, intersect at a single switch point. Figures 11 and 12 show the wage and rent curves, respectively. This fluke case is associated with the disappearance of a range of the scale factor for the rate of profits in no technique is cost-minimizing even though both Alpha and Epsilon have a positive scale factor, and Epsilon has positive rent. It is also associated with the appearance of a range of the scale factor in which both Alpa and Delta are cost-minimizing.

Figure 11: Wage Curves for Third Fluke Case

Figure 12: Rent Curves for Third Fluke Case

5.4 Switch Point at Minimum Scale Factor for Delta

In the penultimate fluke case, Alpha and Epsilon have a switch point at the minimum scale factor for the rate of profits (Figures 13 and 14). This fluke case is associated with the disappearance of the range of the rate of profits at which both Epsilon and Delta are cost-minimizing.

Figure 13: Wage Curves for Fourth Fluke Case

Figure 14: Rent Curves for Fourth Fluke Case

5.5 Alpha vs. Delta Switch Point at Zero Wage

In the last fluke case (Fitures 15 and 16), Delta is only cost-minimizing at the switch point with Alpha. For a smaller scale factor, Delta does not have a non-negative rate of profits. For a larger scale factor, rent under Delta would be negative. This last fluke case is associated with the disappearance of a range of the scale factor for the rate of profits in which both Alpha and Delta are cost-minimizing.

Figure 15: Wage Curves for Fifth Fluke Case

Figure 16: Rent Curves for Fifth Fluke Case

6.0 All Markups in Agriculture

The above has briefly justified the vertical partitions in Figure 17, which shows the variation in the analysis of the cost-minimizing technique with perturbations of the markup up in agriculture. Table 3 shows how the analysis of the choice of technique varies among the numbered regions. If wants to look at these results in some detail, one can relate the variation in the analysis of the choice of technique to the fluke cases. This example demonstrates that my visualization techniques and perturbation analysis can be applied to an example where the cost-mninimizing technique is not found from a frontier of wage curves. The non-uniqueness and non-existence of a cost-minimizing technique arises in D'Agata's original example.

Figure 17: Variation of the Technique with the Markup in Agriculture

Table 3: The Cost-Minimizing Technique in Selected Regions in Parameter Space

Region	Range for Scale Factor	Cost-Minimizing Techniques
1	0 ≤ r ≤ Rδ	Epsilon
	Rδ ≤ r ≤ r*	Delta and Epsilon
	r* ≤ r ≤ Rε	None. Wage for Alpha, Delta, Epsilon positive. Rent for Delta and Epsilon positive.
	Rε ≤ r ≤ R*,δ	None. Wage for Alpha, Delta, positive. Rent for Delta positive.
	R*,δ ≤ r < Rα	None. Wage for Alpha positive.
2	0 ≤ r ≤ Rδ	Epsilon
	Rδ ≤ r ≤ r*	Delta and Epsilon
	r* ≤ r ≤ R*,δ	None. Wage for Alpha, Delta, Epsilon positive. Rent for Delta and Epsilon positive.
	R*,δ ≤ r ≤ Rε	None. Wage for Alpha, Epsilon positive. Rent for Epsilon positive.
	Rε ≤ r < Rα	None. Wage for Alpha positive.
3	0 ≤ r ≤ Rδ	Epsilon
	Rδ ≤ r ≤ r*	Delta and Epsilon
	r* ≤ r ≤ R*,δ	None. Wage for Alpha, Delta, Epsilon positive. Rent for Delta and Epsilon positive.
	R*,δ ≤ r < R*,ε	None. Wage for Alpha and Epsilon positive. Rent for Epsilon positive.
	R*,ε ≤ r < Rα	None. Wage for Alpha positive.
4	0 ≤ r ≤ Rδ	Epsilon
	Rδ ≤ r ≤ R*,ε	Delta and Epsilon
	R*,ε ≤ r ≤ R*,δ	Alpha and Delta
	R*,δ ≤ r < Rα	None. Wage for Alpha positive.
5	0 ≤ r ≤ R*,ε	Epsilon
	R*,ε ≤ r ≤ Rδ	Alpha
	Rδ ≤ r ≤ R*,δ	Alpha and Delta
	R*,δ ≤ r < Rα	None. Wage for Alpha positive.
6	0 ≤ r ≤ R*,ε	Epsilon
	R*,ε ≤ r ≤ Rα	Alpha

7.0 Conclusion

The analysis of the choice of technique with rent, including intensive rent, is more complicated than such analysis in a model with only circulating capital:

"The complexity of the outcomes with the potential existence of conflict or concordance among the three major economic categories (earners of wages, profits, and rents) profoundly modifies the traditional analysis of profits and wages." -- Alberto Quadrio Curzio and Fausta Pellizzari (2010).

The quantity and price systems are interconnected. Assumptions on the level of net output are required to determine which techniques are feasible. Introducing relative market power among industries further complicates the analysis of the choice of technique.