Structural Economic Dynamics and Fake Switch Points

Figure 1: A Pattern Diagram with Joint Production

1.0 Introduction

This post completes an example. I analyzed bits of this example here and here. This post may make no sense if you have not read a long series of previous posts or, maybe, the papers highlighted here and here. I am interested in how and if my approach to analyzing and visualizing variations in the choice of technique with technical progress extends to joint production. The example suggests fake switch points do not pose an insurmountable obstacle for such an extension.

2.0 Technology

I repeat the specification of technology.

I postulate an economy in which two commodities, corn and linen, can be produced from inputs of corn, linen, and labor. Managers of firms know of three processes (Tables 1 and 2) to produce corn and linen. Each process produces net outputs of corn and linen as a joint product. Inputs and outputs are specified in physical units (say, bushels and square meters) per unit level of operation of the given process. Inputs are acquired at the start of the year, and outputs are available for sale at the end of the year.

Table 1: Inputs for The Technology

Input	Process
	(a)	(b)	(c)
Labor	eσ0,1(1 - t)	eσ0,2(1 - t)	eσ0,3(1 - t)
Corn	20	20	30
Linen	20	20	30

Table 2: Outputs for The Technology

Output	Process
	(a)	(b)	(c)
Corn	21	23	36
Linen	27	25	34

I assume that requirements for use are such that two processes must be operated to satisfy those requirements. I need to investigate the implications of this assumption further. Apparently, for this example, it implies that the economy is not on a golden rule steady state growth path, with the rate of profits equal to the rate of growth. Anyway, with this assumption, three techniques - Alpha, Beta, and Gamma - can be operated. Table 3 specifies which processes are operated for each technique.

Table 3: Techniques

Techniques	Processes
Alpha	a, b
Beta	a, c
Gamma	b, c

The technology, as I have defined it, is parameterized. I consider the following specification for the rate of decrease in labor coefficients.

σ_0,1 = 2

σ_0,2 = σ_0,3 = 5/2

Bidard & Klimovsky's example arises when t is unity.

3.0 Prices and the Choice of Technique

A system of two price equations arises, for each technique. I assume the labor coefficient is treated as a constant over the period of production - say, a year. With linen as numeraire, these equations for the Alpha technique are:

(20 p₁ + 20)(1 + r) + [e^{σ_0,1(1 - t)}] w = 21 p₁ + 27

(20 p₁ + 20)(1 + r) + [e^{σ_0,2(1 - t)}] w = 23 p₁ + 25

One can these equations for two variables in terms of, say, the rate of profits. For each technique, its wage curve shows the wage as a function of the rate of profits. One cannot generally base the choice of technique, under joint production, on figuring out which technique contributes to the outer frontier at a given rate of profits.

Instead, one can calculate profits and losses, with the given rate of profits and a technique's price system for the processes not in the technique. This exercise only makes sense when the rate of profits, the wage, and prices are non-negative for the starting technique. The technique is cost-minimizing only if no extra profits can be made with processes outside the technique.

I deliberately frame this as a combinatorial argument. Bidard likes what he calls a market algorithm, where, when one identifies a process earning extra profits, one introduces the process into the technique. In the case of joint production, it is not clear which process should be dropped. Furthermore, examples exist in which a cost-minimizing technique exists but cannot be reached from certain starting points with the market algorithm.

4.0 Patterns

I have constructed the figure at the top of the post to illustrate how the choice of technique varies with technical progress in this example. The dashed lines highlight features of the example that do not bear on the choice of technique. The light vertical solid lines divide time into numbered regions. Table 3 lists the cost-minimizing techniques, in order of an increasing rate of profits in each region.

Table 3: Regions

Regions	Techniques
1	Gamma, No Production, Alpha
2	Gamma, No Production, Alpha
3	Gamma, Alpha & Gamma, Alpha
4	Alpha & Gamma, Alpha
5	Beta, Alpha & Gamma, Alpha

I could say a lot more about the example. I will note that in region 1, the wage increases with the rate of profits, for the Alpha technique, in the interval for the rate of profits where both wages and the price of corn are positive. In region 2, the wage decreases with the rate of profits, for the Alpha technique. The division between regions 2 and 3 is associated with that interval for the rate of profits for Alpha transitioning to have a non-empty intersection with the similar interval for the Gamma technique. for

5.0 Conclusion

This post has illustrated that one type of my types of pattern diagrams can apply to joint production. This type illustrates how the relationship between the choice of technique and distribution varies with technical progress. It can be constructed even in cases, such as joint production, where the choice of technique cannot necessarily be based on wage-rate of profits curves and their outer frontier.

If fake switch points are not shown, this type of pattern diagram does not depend on the specification of the numeraire. If the ordinate in Figure 1 were the wage, instead of the rate of profits, it would be upside down, in some sense. A different numeraire would rescale the wage. When corn is numeraire, only one fake switch point exists. It, too, would be a horizontal line segment. But fake switch points are fake precisely because they do not impact the choice of technique. They can be left off the diagram.

The example also illustrates new types of patterns for dividing adjacent regions. Under joint production, a technique can be associated with non-negative prices and a wage for an interval of the rate of profits that does not include a rate of profits of zero. Both the Alpha and the Beta technique exhibit this possibility in the example. And we can divide regions based on when the range of rate of profits in which such a technique becomes cost-minimizing comes to include zero or begins to interact with the range in which another technique is cost-minimizing

This example also illustrates that the cost-minimizing technique may not be unique in a range of rates of profits. I think this non-uniqueness is qualitatively different than how non-uniqueness can arise in models with only circulating capital. In circulating capital models, non-uniqueness is associated with two techniques having identical wage curves. Not so here.

I do not intend to write this example up any more extensively. I have no so-called paradoxical behavior here, such as reswitching, reverse capital-deepening, or the reverse substitution of labor. I may go on to explore where techniques are described by rectangular matrices, with more produced commodities than processes, and there is a dependence on the requirements for use.

References

• Bidard, Christian and Edith Klimovsky (2004). Switches and fake switches in methods of production. Cambridge Journal of Economics. 28 (1): 89-97.