A Disconcerting Example of Intensive Rent From D'Agata

Figure 1: The Wage Frontier And Rent

1.0 Introduction

This post is another worked homework example, problem 7.8 in Chapter 10 of Kurz and Salvadori (1995). The example illustrates the possible non-existence of a cost-minimizing technique with intensive rent. I once looked at an example from J. E. Woods of joint production. I claim that that example does not make the desired point, given the possibility of a price of zero for some produced good. I do not think this example of rent can be resolved like that.

Kurz and Salvadori suggest to me how I might apply my perturbation techniques: "...calculate what will happen if either only process (4) or only process (5) were missing."

2.0 Technology, Techniques, and Requirements for Use

Anyways, Table 1 presents the available technology. Corn is grown on homogeneous land, and three processes are available for producing corn. One hundred acres of land are available, leading to the possibility of two processes being operated side-by-side with positive rent.

Table 1: The Coefficients of Production

Input	Industries and Processes
	Iron	Steel	Corn
	I	II	III	IV	V
Labor	1	1	1	11/5	1
Land	0	0	1	1	1
Iron	0	0	1/10	1/10	1/10
Steel	0	0	2/5	1/10	1/10
Corn	1/10	3/5	1/10	3/10	2/5

Table 2 shows the processes operated in each of the six techniques available. (All three corn-producing processes are operated only at a switch point where the Delta, Epsilon, and Zeta techniques are simultaneously cost-minimizing. Iron, steel, and corn are basic commodities in all techniques. Land is never a basic commodity.

Table 2: Techniques

Technique	Process
Alpha	I, II, III
Beta	I, II, IV
Gamma	I, II, V
Delta	I, II, III, IV
Epsilon	I, II, III, V
Zea	I, II, IV, V

Requirements for use are 90 tons iron, 60 tons steel, and 19 bushels corn. Alpha, Delta, and Epsilon can meet requirements for use. That is, one can find levels of operation of the processes comprising these techniques such that the net output of the economy is the previously specified vector and no more than 100 acres of land are farmed. Beta, Gamma, and Zeta are infeasible.

3.0 Cost-Minimizing Techniques and the Wage Frontier

When Alpha is used, not all land is farmed. Rent would be zero. But Alpha is never cost-minimizing.

Figure 2: Extra Profits At Delta and Epsilon Prices

To see if a technique is cost-minimizing at a given rate of profits, find prices of production, the wage, and rent for the technique. Then one can calculate extra profits for every process. Costs include the going rate of profits on advances for purchasing capital goods, wages, and rents. Figure 2 plots extra profits for processes for Delta and Epsilon prices.

The left panel illustrates Delta. No extra profits are made or extra costs are incurred in processes I, II, III, and IV. Delta only has a non-negative wage and a non-negative rent between a rate of profits of 1/9 (that is, approximately 11.1 percent) and approximately 52.3 percent. From a rate of profits of approximate 11 percent to 46 percent, extra profits cannot be made in operating process V. Delta is cost-minimizing.

For a higher rate of profits, in a range in which rent is non-negative under Delta prices, process V makes extra profits. Delta is not cost-minimizing. Which technique would be adopted under these conditions? Process V could be be the only corn-producing process, in the Gamma technique. But that technique is not feasible. Suppose process V replaces process III, in the Zeta process. That technique results in more being produced than are needed for requirements for use. Epsilon is the only feasible technique in which land is fully farmed and two corn-producing processes are operated, with a positive rent.

The right panel in Figure 2 illustrates extra profits for all processes under Epsilon prices. Epsilon has a non-negative wage and a positive rent up to a rate of profits of 2/3 (that is, approximately 66.7 percent) In the range of the rate profits from zero to approximately 46 percent, Epsilon is cost-minimizing. For a higher rate of profit, where the wage is still non-negative under Epsilon, process IV makes extra profits. I highlight in this range when Delta is feasible and consistent with a positive wage and positive rent.

The above analysis shows how the wage frontier is constructed in this example. The wage frontier is illustrated in the left panel in Figure 1 at the top of this post. The corresponding rent is shown in the right panel. A range of rate of profits exists in which Delta and Epsilon are both cost-minimizing. The switch point between Delta and Epsilon is at 19/41, (that is, approximately 46.3 percent). Above this rate of profits, no technique is cost-minimizing.

4.0 Conclusion

Between rates of profits of 19/41 and approximately 52.3 percent, Epsilon makes extra profits at Delta prices, and Delta makes extra profits at Epsilon prices. Even though feasible techniques exist that are consistent with positive wages, rates of profits, rent, and prices of production, no cost-minimizing technique need exist.

References

• D'Agata, Antonio. 1983a. The existence and unicity of cost-minimizing systems in intensive rent theory. Metroeconomica 35: 147-158.
• Kurz, Heinz D. and Neri Salvadori. 1995. Theory of Production: A Long-Period Analysis. Cambridge: Cambridge University Press.