Bidard And Klimovsky's Fake Switch Point

Figure 1: Wage Curves with A Fake Switch Point

1.0 Introduction

I find that, in developing an example on fixed capital and extensive rent, I need to review fake switch points. I am relying on Bidard's book, not on his paper with Klimovsky.

In this post, I just run through their example. They should have been more explicit about which techniques are feasible in their exposition.

2.0 Technology

Bidard and Klimvsky's example is one of pure joint production. The technology consists of the three processes shown in Table 1. Each column shows the inputs of person-years of labor, bushels of corn, and number of hogs needed as inputs for a unit level of operation. The outputs of bushels corn and number of hogs are also shown. Since each process has an output of more than one commodity, this technology is one of joint production. I assume constant returns to scale, and that each process takes a year to complete.

Table 1: Three Processes Comprise the Technology

Input	Processes
	Process 1	Process 2	Process 3
Labor	a0,1 = 1	a0,2 = 1	a0,3 = 1
Corn	a1,1 = 20	a1,2 = 20	a1,3 = 30
Hogs	a2,1 = 20	a2,2 = 20	a2,3 = 30
Outputs
Corn	b1,1 = 21	b1,2 = 23	b1,3 = 36
Hogs	b2,1 = 27	b2,5 = 20	b2,3 = 34

Three techniques (Table 2) can be constructed from these processes, when neither corn or hogs are a free good. Unlike in single production, a single technique cannot produce any composition of net output.

Table 2: Techniques of Production

Technique	Processes
Alpha	1, 2
Beta	1, 3
Gamma	2, 3

Another issue arises in the choice of technique that does not occur with joint production. Suppose the processes in Alpha are being operated. And suppose extra profits can be obtained with the the third process at Alpha prices. Which process in Alpha should be replaced? If the first process is replaced, the Gamma technique is adopted. If the second process is replaced, the Beta process is adopted. In single production, each process is associated with a specific industry, producing a specific commodity. In this example of joint production, considerations of feasibility are sufficient to answer this question.

3.0 Requirements for Use, Prices and the Choice of Technique

I consider two cases. The composition of net output is specified in different proportions in the two cases. Two techniques, out of three, are feasible in each case. But which technique is infeasible varies.

A price system is associated with each technique. Each process in the technique yields an equation. I assume that wages are paid out of the surplus at the end of the year. These equations for prices of production show the same rate of profits being made in both operated processes. I take net output as the numeraire. A third and final equation specifies that the price of the numeraire is unity.

The price equations can be solved, given the rate of profits. The wage, the price of corn, and the price of hogs are each a function of the rate of profits for each technique.

The question posed by the example is whether the outer frontier of the wage curves corresponds to the cost-minimizing technique. It does not. But, in the example when it does not, the wage curve on the outer frontier is not feasible.

3.1 Case 1

Suppose the requirements for use, that is, net output consists of 0.38 bushels corn and 0.62 hogs. Then Alpha is infeasible, and Beta and Gamma are feasible.

Two fluke cases are of interest here. If net output consists of 3/8 bushels corn and 5/8 hogs, all three techniques are feasible. Only the second processes are operated for the Alpha and Gamma techniques. The first and third processes are operated at positive levels for Beta. On the other had, a net output of 3/5 bushels corn and 2/5 hogs is an edge case where Alpha is infeasible, with Beta and Gamma feasible. Only the third process is operated for Beta and Gamma.

For the remainder of this section, I consider the specific intermediate case where Alpha is infeasible. Figure 2 graphs the wage curves. Only one switch point exists, at a rate of profits of 10 percent. Figure 3 shows extra profits for each profits at Beta prices. Beta is cost-minimizing for a non-negative rate of profits up to 10 percent. After that, extra profits can be obtained by operating the second process at Alpha prices.

Figure 2: Wage Curves with Alpha Infeasible

Figure 3: Case 1, Extra Profits at Beta Prices

Figure 4 shows extra profits for each process at Gamma prices. Gamma is cost-minimizing from the switch point at a rate of profits of 10 percent to the maximum rate of profits. For rates of profits smaller than in this rate, extra profits can be obtained by operating process 1.

Figure 4: Case 1, Extra Profits at Gamma Prices

In this case, the wage curve on the outer envelope for a rate of profits between 10 percent and the maximum rate of profits of 18 percent is not the wage curve for the cost-minimizing technique. But it is a wage curve, Gamma, for an infeasible techinque. If only wage curves for feasible techniques were considered, the construction of the outer frontier of wage curves would be a correct analysis of the choice of technique.

3.1 Case 2

Suppose now that net output consists of 0.13 bushels corn and 0.87 hogs. Alpha and Beta are feasible. Gamma is infeasible. (Here a fluke case is a net output of 1/8 bushels corn and 7/8 hogs. Alpha and Beta remain feasible, but only the first process is operated in each case at a positive level. Processes 2 and 3 are operated at a level of 0.)

Figure 1, at the top of this post, shows the wage curves. In this case, the wage curves on the outer frontier intersect at two points, the switch point at 10 percent and the fake switch point at approximately 13 percent.

The price system is identical among techniques at a switch point. Figure 5 shows how the price of corn varies with the rate of profits for the techniques in this case. Figure 6 shows how the price of hogs vary. You can see the switch point on these graphs of prices. No manifestation of the fake switch point is apparent.

Figure 5: Case 2, Price of Corn

Figure 6: Case 2, Price of Hogs

You can graph extra profits for each process, at prices for each technique, here too. Figure 7 graphs extra profits for Alpha. Alpha is cost-minimizing from the switch point at a rate of profits of 10 percent to the maximum at 20 percent. And Figure 8 demonstrates that Beta remains cost-minimizing for rates of profits below 10 percent.

Figure 7: Case 2, Extra Profits at Alpha Prices

Figure 6: Case 2, Extra Profits at Beta Prices

So Bidard and Klimovsky's claim that an intersection of wage curves on the outer frontier can be a fake switch point. In their example, one of the wage curves at the fake switch point is for an infeasible technique. And when the wage curve for the cost-minimizing technique lies below the outer frontier, the curve on the frontier is for that infeasible technique.

4.0 Conclusion

This example illustrates the interaction between requirements for use and the feasibility of techniques of production in models of joint production. The appearance of the wage curves depends on the numeraire. Bidard does not tie the numeraire to the requirements for use, as I do. He considers numeraires (one bushel corn for one case and one hog for the other) which cannot be feasibly produced with any technique. Likewise, Bidard does not consider feasibility. He discusses the angles between certain vectors in quantity space. I do not understand this without more studying, but perhaps it is a matter of feasibility.

Anyways, the example demonstrates the fallacy of analyzing the choice of technique by the construction of the outer wage frontier. In the first case, the cost-minimizing technique is not on the outer frontier for large rates of profits. The technique on the outer frontier is infeasible in this range.

The wage curve for the cost-minimizing technique is not always on the outer frontier in the second case either. Here, too the wage curve on the outer frontier is for a technique infeasible in this range. Furthermore, a fake switch point appears. The wage curves on the outer frontier intersect at a point where the cost-minimizing technique does not change and is unique. The prices of corn and hogs vary with the technique at the fake switch point. This variation is not possible in the single production models.

Which technique is cost-minimizing at a given rate of profits does not depend on the numeraire when the numeraire is disassociated with the net product.

I did not consider techniques in which only one process is operated to satisfy requirements for use. Perhaps there are such techniques in which one good is overproduced and becomes a free good.

References

• Bidard, Christian. 2004. Prices, Reproduction, Scarcity. Cambridge: Cambridge University Press.
• Bidard, Christian and Edith Klimovsky. 2004. Switches and fake switches in methods of production. Cambridge Journal of Economics 28 (1): 88-97.