Perturbations Of Selected Parameters In The Corn-Tractor Model

Figure 1: Partitioning a Part of the Parameter Space with Fluke Cases

1.0 Introduction

This post is a continuation of the first example here. I examined a perturbation of two parameters of that example. I ended up with a more perspicacious partition of the parameter space than here.

2.0 Technology

Table 1 merely repeats the parameters for the fluke case that I started with. This case has switch points on the axis for the rate of profits and on the wage axis. A third switch point exists at an intermediate rate of profits.

Table 1: Parameters for Technology for First Example

Parameter	Type I Tractors	Type II Tractors
Tractor input per tractor (a)	≈ 0.306226	2/5
Labor input per tractor (b)	≈ 233.6967	20
Years tractors last in tractor industry (n)	1	2
Tractor input per bushel corn (α)	1	20
Labor input per bushel corn (β)	αI bI/aI	850
Years tractors last in corn industry (ν)	1	2

3.0 Perturbations of Selected Parameters

Almost any perturbation of the model parameters destroys fluke properties of the example in the previous section. Figure 1 illustrates perturbations in a_II and α_II. A switch point is on the axis for the rate of profits only for a specific value of a_II. Likewise, a switch point is on the wage axis only for the depicted partition of the parameter space, of, for instance, regions 1 and 7. The example in the previous post has parameters found at the intersection of these two partitions. The two other partitions occur for parameter values at which a switch point is repeated and the two wage curves are tangent at this switch point. The regions bounded by these partitions of the selected part of the parameter space are numbered.

The dashed line depicts the combination of coefficients of production for which the ratio of labor to tractors does not vary between industries for tractors of type II. To the left and above this line the physical capital-intensity of production is less, for type II tractors, in producing new tractors than it is in producing corn. To the right and below, the tractor industry for type II tractors has a larger physical capital-intensity than corn production.

I would have liked to have drawn the partitions as three-dimensional manifolds in a four-dimensional space, where (a_II, b_II, α_II, β_II) is a point in the space. But I can visualize a tesseract only momentarily, if at all (Heinlein 1941). Figure 1 is constructed by selecting only two parameters to perturb.

Double-fluke cases arise at intersections of the partitions. The partition between regions 2 and 3 is tangent to the partition between regions 3 and 4 at their point of intersection. Similarly, the partition between regions 1 and 5 is tangent to the partition between regions 2 and 5. The two partitions between regions 6 and 7 are tangent at their point of intersection, as well.

This last double-fluke switch point can perhaps admit of some elaboration. Figure 2 shows the rate of profits and the wage at switch points for each of two fluke cases. The solid lines correspond to the partition between regions 1 and 5 and the lower partition in Figure 2 between regions 6 and 7. The dashed lines correspond to the partition between regions 3 and 4 and the upper partition. Three switch points exist for the parameters along these two partitions. Two of these switch points are repeated roots, which is the fluke case under consideration. All three switch points coincide on the wage frontier at the double-fluke switch point to the extreme left.

Figure 2: The Rate of Profits at Switch Points with Tangent Wage Curves

Table 2: Cost-Minimizing Techniques by Region

Region	Cost-Minimizing Technique	Notes
1	Type II	No switch point. Type II tractors are dominant with sufficiently low coefficients of production in producing Type II tractors.
2	Type II, Type I	Around the switch point, a lower rate of profits is associated with a more roundabout technique, a greater capital-output ratio, and more consumption per person-year.
3	Type I, Type II, Type I	Around the first switch point, a lower rate of profits is associated with a less roundabout technique, a higher capital-output ratio, and more consumption per person-year. Around the second switch point, a lower rate of profits is associated with a more roundabout technique, a lower capital-output ratio, and less consumption per person-year.
4	Type I	No switch point. Type I tractors are dominant with sufficiently high coefficients of production in producing Type II tractors.
5	Type II, Type I, Type II	Around the first switch point, a lower rate of profits is associated with a more roundabout technique, a greater capital-output ratio, and more consumption per person-year. Around the second switch point, a lower rate of profits is associated with a less roundabout technique, a lower capital-output ratio, and less consumption per person-year.
6	Type I, Type II, Type I, Type II	Around the first and third switch point, a lower rate of profits is associated with a less roundabout technique, a greater capital-output ratio, and more consumption per person-year. Around the second switch point, a lower rate of profits is associated with a more roundabout technique, a smaller capital-output ratio, and less consumption per person-year.
7	Type I, Type II	Around the switch point, a lower rate of profits is associated with a less roundabout technique, a greater capital-output ratio, and less consumption per person-year.

The analysis of the choice of technique is qualitatively invariant in each numbered region. Table 2 lists the cost-minimizing technique, in order of an increasing rate of profits, in each region. One technique is cost-minimizing, whatever the distribution of income, in regions 1 and 4. One switch point exists in regions 2 and 7. Double-switching occurs in regions 3 and 5. Finally, triple-switching occurs in region 6. Perturbations of the parameters for an example in the previous post can result in each type of tractor being cost-minimizing in two discrete ranges of the wage or the rate of profits. This partitioning is not unique to this model.

4.0 Conclusions

I can examine specific properties of the switch points in each region, and maybe draw some more conclusions. But that will be for future posts.