Variation On An Example From Schefold

Figure 1: Variation in the Economic Life of a Machine with Technical Progress

This post varies the coefficients of production in an example from Bertram Schefold. I wanted to have 'nice' fractions at a time of zero. Qualitatively, this looks like a previous post.

Reviewers for a recent rejection of an article with another fixed capital example objected to this type of model. I need to relate technical progress to a well-known type (Harrod-neutral, Marx-biased, or whatever) or produce some evidence that this sort of modeling is reasonable.

Table 1 presents the technology for this example. Machines and corn are produced in this economy. Corn is the only consumption good. New machines are produced from inputs of labor and corn. Corn is produced from inputs of labor, corn, and machines. A machine can be worked for two years. After the end of the first year of its working life, it is known as an old machine. I assume each process requires a year to complete and exhibits constant returns to scale.

Table 1: Coefficients of Production for The Technology

Input	Process
	(I)	(II)	(III)
Labor	(7/25) e- σ t	3 e- φ t	(14/5) e- φ t
Corn	(4/25) e- σ t	(4/25) e- φ t	(2/3) e- φ t
New Machines	0	1	0
Old Machines	0	0	1
Outputs
Corn	0	1	1
New Machines	1	0	0
Old Machines	0	1	0

The managers of firms need not run the machine for two years. They could discard the machine after only one year. (I assume free disposal.) I call Alpha the technique in which the machine is run for one year, and Beta the technique in which the machine is run for its full physical year of two years. The managers will be cost-minimizing if they run the machine for only one year if the price of an old machine is negative. Prices are found as prices of production, with an external specification of the wage or the rate of profits.

Figure 1 above and Table 2 below illustrate the analysis of the choice of technique. Until time reaches the pattern over the axis for the rate of profits, it is cost-minimizing to operate the machine for only one year. In Region 2, the machine is operated for two years when wages are low, and for one year when wages are higher. Region 3 is an example of reswitching. Eventually, it is cost-minimizing to operate the machine for two years, for all feasible wages.

Table 2: The Wage Frontier in Selected Regions in Parameter Space

Region	Switch Points	Cost-Minimizing Techniques
1	None	Alpha
2	One	Beta, Alpha
3	Two	Beta, Alpha, Beta
4	None	Beta
5	One	Alpha, Beta

Figure 2 partitions the parameter space such that the characteristics of the variations in the choice of technique technique do not vary within each region. Each region is bounded by thick (non-dotted) lines. Suppose that technical progress in producing and using machines is steady. That is, σ and φ have some fixed values. The dotted line in Figure 2 illustrates a path in logical time for such a thought experiment. The 45 degree line corresponds to the case in which σ and φ are equal, as in Figure 1. The numbering of the regions in Figures 1 and 2 and in Table 2 correspond.

Figure 2: A Parameter Space

Stepping through this example does not provide any qualitative differences from previous posts. Reswitching in models of fixed capital can be manifested as the return of a period of truncation. Around a switch point, a lower wage need not be associated with greater employment for a given net output. And so on.