Pattern Analysis for a Fixed Capital Example

Figure 1: A Pattern Diagram

1.0 Introduction

In this example, I perturb parameters in an example of Bertram Schefold's. I was disappointed in that, as far as I can see, one can analyze the choice of technique in this example by the construction of the wage-rate of profits frontier. As far as I understand, this is not true for joint production in general. I guess I also need to find an example in which the physical life of a machine is at least three years so as to find a three-technique pattern.

This example does highlight differences in different measures of capital-intensity.

2.0 Technology

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

Inputs	Industry
	Machine	Corn
Labor	a0,1 = (1/10) u(t)	a0,2 = (43/40) u(t)	a0,3 = u(t)
Corn	a1,1 = (1/16) u(t)	a1,2 = (1/16) u(t)	a1,3 = (1/4) u(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

I model technical progress by constantly decreasing inputs into each process, other than machines:

u(t) = e^{1 - σ t}

When σ t is unity, this is Bertram Schefold's example of reswitching, at rates of profits of 1/3 and 1/2.

3.0 Prices of Production

The first row in Table 1 can be summarized by a row vector, a₀, of labor coefficients. The next three rows are expressed by a square matrix A. The last three rows form the matrix B. Suppose wages are paid out of the surplus product at the end of the year. If the same rate of profits is to be made in all operating processes, prices must satisfy the following system of equations;

p A (1 + r) + w a₀ = p B

I let corn be the numerator:

p e₁ = 1

where e₁ is the first column of the identity matrix.

Given the wage, w, in a range between zero and some maximum, the above system of price equations can be solved for the rate of profits, r, the price of a new machine, p₂, and the price of an old machine, p₃.

4.0 Choice of Technique

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.) The managers will be cost-minimizing if they run the machine for only one year if the price of an old machine is negative.

Alternatively, consider the price system when the machine is operated only two years. The matrices A and B are 2x2 square matrices, and a₀ is a row vector with two elements. With these prices and the price of an old machine of zero, one could calculate the cost of operating the machine for a second year to produce a bushel of corn. When this cost is less than unity (the price of a bushel of corn), it is cost-minimizing to operate the machine for both years.

These two methods of analyzing the choice of technique yield the same answer for this example. Figure 1, above, illustrates the results. 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.

5.0 Capital

In outdated neoclassical intuition, a higher wage indicates that labor is more scarce, in some sense, and capital is relatively more abundant. One might, wrongly, except the price system to encourage capitalists to adopt less labor-intensive or more capital-intensive techniques, in some sense. And, in a simple example like this one, one might expect the more capital-intensive technique to be one in which the machine is run for both years.

The example confounds these expectations in both Region 2 and Region 3. Around the switch point in Region 2, a higher wage is associated with the adoption of a technique in which the machine is only operated for the first year. The same is true of the same switch point - the one at the lower wage - in Region 3. From this viewpoint, the switch point is "perverse" in both regions.

This result contrasts with the usual analysis based on real Wicksell effects. The real Wicksell effect is negative for the switch point in Region 2. It is positive for the same switch point in Region 3. For a switch point with a negative real Wicksell effect, a higher wage is associated with the adoption of a technique with more net output per person-year employed. And that is so in this case too. The switch point is only 'perverse', from this perspective, in Region 3.

6.0 Conclusion

This post has illustrated that what I am calling pattern analysis can be applied to examples of joint production in which joint production is only manifested in production and use of long-lived machines. It has focused attention on the distinction between different intuitions about the capital-intensity of a technique.