Friday, December 16, 2016

Perturbation Of A Reswitching Example

Figure 1: Wage-Rate of Profits Curve for Two Techniques
1.0 Introduction

In this post, I consider a perturbation of the data on technology in this example of the production of commodities by means of commodities. This example is of the choice of technique from two techniques. Each technique can be used to produced a commodity, corn, used for consumption and as the numeraire. The perturbations considered here drastically changes the qualitative characterization of the technology. And they only slightly change the location of switch points and the maximum wages, for the two techniques. These perturbations also only slightly change the maximum rate of profits for one technique. They do, however, drastically lower the maximum rate of profits for the other technique.

2.0 Two Techniques With Two Perturbations

Table 1 displays the technology available to the firms in this example. (I have renamed the industries and commodities.) Each column defines the coefficients of production for a process for producing the output of an industry. Only one process is known for producing iron, and only one process is available for producing steel. Two processes are known for producing corn. Coefficients of production show how much of each input must be available, to provide flows of services of that input over the year, per unit output produced and available at the end of the year. The parameters δ and ε must both be nonnegative for a given technology.

Table 1: The Technology
Corn Industry
Labor (Person-Years):1/31/213/2
Iron (Tons):1/6ε10
Steel (Tons):ε1/401/4
Corn (Bushels):δδ00
Output (Various):1111

Two techniques are defined, in this technology, for producing a net output of corn. Each technique consists of a single process for producing corn and whichever of the iron-producing and steel-producing processes (sometimes both) is needed to reproduce the capital goods used up in producing a net output of corn.

2.1 No Basic Commodities

Consider the special case where:

δ = ε = 0

In this case, one can say that in both techniques, no commodity is basic. Or one might say that, in each technique, one commodity is basic, and that which commodity is basic varies with the technique. It depends on how you look at it.

In the Alpha technique, corn is produced with the process labeled Alpha. Iron is used as an input in producing iron and in producing corn. Corn is not an input in any process, and steel is not produced. If one disregarded the non-produced commodity, steel, one could say iron is the single basic commodity. On the other hand, if one included steel as a possible commodity, iron would not be basic, since it does not enter into the production of steel, either directly or indirectly.

The same paragraph could be written about the Beta technique, with the role of iron and steel reversed.

2.2 Three Basic Commodities

Cosider a case in which both the δ and ε parameters are (small) positive numbers. I worked out the following case:

δ = 1/300
ε = 1/200

In this case, the Alpha technique consists of the iron-producing process, the steel-producing process, and the corn-producing process labeled Alpha. All three commodities are basic. Corn enters indirectly into the production of corn through both iron and steel. Similarly, all three commodities are basic in the Beta technique.

So one sees that the structure of production, in both techniques, is qualitatively different in these two cases. This difference is seen in which commodities are basic, and which are not.

3.0 Wage-Rate of Profits Curves

The managers of firms choose the processes comprising the technique so as to minimize cost. Let a bushel of corn be the numeraire. Suppose labor is advanced, and wages are paid out of the output available at the end of the year.

For each technique, these assumptions are such that a relation between the wage and the rate of of profits arises. Both the wage and the rate of profits range between zero and a finite maximum wage or rate of profits. The higher the rate of profits, the lower the wage and vice versa. You can see these wage-rate of profits curves graphed in the first figure here for the special case in which δ = ε = 0. Figure 1, at the top of this post, graphs these wage curves for the specific positive values of δ and ε graphed above.

The choice of technique can be analyzed based on the outer frontier of the wage-rate of profits curves for the technique. For a given rate of profits, the cost-minimizing technique is the one with the highest wage at that rate of profits. At switch points, more than one technique is cost-minimizing. Firms can adopt a linear combination of the techniques on the outer frontier at switch points.

This is a reswitching example for the perturbations considered here. The Alpha technique is cost-minimizing at low and high rates of profits, with the Beta technique cost-minimizing at intermediate rates of profits. Tables 1 and 2 specify the location of the switch points, as well as the maximum wages and rates of profits for the two techniques. These solution values can be found as easy-to-calculate rational numbers for the original case, as shown in Table 1. Table 2 lists approximate values.

Table 1: The Model with δ = ε = 0
VariableAlpha TechniqueBeta Technique
Maximum Wage5/7 = 0.71433/5 = 0.6
Maximum Rate of Profits500%300%
First Switch Point
Wage1/2 = 0.5
Rate of Profits100%
Second Switch Point
Wage1/3 = 0.3333
Rate of Profits200%

Table 1: Perturbed Model
VariableAlpha TechniqueBeta Technique
Maximum Wage0.70940.5991
Maximum Rate of Profits298.0%294.1%
First Switch Point
Rate of Profits84.15%
Second Switch Point
Rate of Profits231.2%

Small variations in the data defining the technology results in small variations in, for example, the maximum wages and the location of switch points. Decreased requirements for commodity inputs in production processes results in an outward movement of the wage-rate of profits curves and the outer frontier. But some changes resulting from these perturbations of the data are discontinuous. The maximum rate of profits is the most noticeable in this example. When iron is the only input in the iron-producing process, the maximum rate of profits for Alpha is 500%. (This maximum depends only on how much iron is required to produce a unit output of iron.) A perturbation that results in all three commodities being basic in both techniques abruptly lowers this maximum rate of profits to below 300%, the maximum rate of profits in the Beta technique in the original example. I also like that the perturbed model, with three basic commodities, removes the necessity for the convexity of a wage curve to be fixed in one direction for the entire curve.

4.0 Conclusion

This example has illustrated the transformation of a simple reswitching example, through perturbations, to another example, in which all commodities are basic. In this three-commodity example, with all commodities basic, the wage-rate of profits curve for the Alpha technique varies in convexity along its extent. Such a variation in convexity is a general property of multicommodity models of the production of commodities by means of commodity, but cannot be seen in two-commodity examples.

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