A 1D Diagram For A Triple-Switching Example

Figure 1: Triple Switching with Strucutral Economic Dynamics

1.0 Introduction

I have been using fluke switch points to partition two-dimensional slices of parameter spaces. I know, I think, how reswitching can appear and disappear. But I am confused how more switch points can appear. So this post is a start on exploring a triple-switching example.

I have stumbled upon two examples of triple-switching, so to speak. I have not yet replicated Steedman's claim that triple-switching can arise in his corn-tractor model. But then, in my first explorations I had different types of tractors lasting for the same number of years. So I turn to an example from Schefold, which I have explored previously.

I expect to find numerical examples of phenomena that I have not yet seen. For example, consider a reswitching example in which the first switch point has negative real Wicksell effects, and the second switch point has positive real Wicksell effects. So the first switch point is 'non-perverse' so far. But the first switch point can exibit the reverse substitution of labor. Around the switch point, a higher wage is associated with more employment in the corn industry per (gross) bushel corn produced. This occurs when there is a third switch point between -100 percent and zero.

2.0 Technology

Table 1 presents the structure of coefficients of production for an example. Each column shows the inputs and outputs, in physical quantities, when the process is operated at unit level. All processes exhibit constant returns to scale.

Table 1: Coefficients of Production

Input	Process
	I	II	III	III
Labor	a0, 1	a0, 2	a0, 3	a0, 4
Corn	a1, 1	a1, 2	a1, 3	a1, 4
New Machines	0	0	1	0
Old Machines	0	0	0	1
Output
Corn	0	1	b1, 3	b1, 4
New Machines	1	0	0	0
Old Machines	0	0	1	0

Three techniques (Table 2) are possible with this technology. Under Alpha, labor and corn are used to produce corn directly. No machines are produced. The Beta and Gamma techniques are more roundabout. First, labor and corn are used to build a machine. Labor works with the machine and inputs of corn to produce more corn. Beta and Gamma differ in whether or not the machine is run for its full physical life of two years. The machine is assumed to be able to be costlessly discarded. Under Beta, the machine is only run for one year.

Table 2: Techniques of Production

Technique	Processes
Alpha	II
Beta	I, III
Gamma	I, III, IV

I make some specific assumptions for the values of coefficients of production:

a_{0, 1} = (3/140) e^{1 - φ t}

a_{0, 2} = e^{(1/2) - σ t}

a_{0, 3} = e^{1 - φ t}

a_{0, 4} = (1/3) e^{1 - φ t}

a_{1, 1} = (31/504) e^{1 - φ t}

a_{1, 2} = (1/2) e^{(1/2) - σ t}

a_{1, 3} = (1/4) e^{1 - φ t}

a_{1, 4} = (2/315) e^{1 - φ t}

b_{1, 3} = 1/2

b_{1, 4} = 1/2

I do not claim that this model of technical change is at all realistic. The idea is to end up with parameters that can be perturbed:

• φ: The rate of decrease of coefficients of production for inputs in processes I, III, and IV.
• σ: The rate of decrease of coefficients of production for inputs in process II.

For given values of φ and σ, productivity improves with time.

3.0 Price Systems

Consider prices of production. I assume that, for the selected technique, the same rate of profits is obtained in all operated processes. Wages are paid out of the surplus at the end of the year. Corn is the numeraire.

Figure 1, at the top of the post, shows the cost-minimizing technique at each level of the wage, for the indicated values of φ and σ. The cost-minimizing technique varies with the illustrated structural economic dynamics. Schefold's example arises at t = 10. The wage frontier is not particularly striking. The difference in the wage curves for Alpha and Gamma are barely distinguishable to the eye.

4.0 Price Conclusion

My next step is to graph (σ t) against (φ t) for various fluke cases. Maybe I will present intermediate results before I figure out what the full parameter space looks like.

So now I have several problems queued up:

1. Explore how triple-switching can arise with partitions of a two-dimensional parameter space.
2. See if I can get publsihed my construction of Hayekian triangles from models of prices of production.
3. Update and see if I can get published my working paper with one-dimesional diagrams.
4. Write up another refutation of Austrian claims. Recall I want to mention that this is only part of a larger demonstration that capital-intensity is not to be explicated in terms of a period of production.
5. Write up an exposition of local perturbations of fluke switch points with two-dimensional diagrams.
6. See if I can make sense of the order of efficiency and the order of rentability in a model that combines intensive and extensive rent.