An Example With Intensive And Extensive Rent

Figure 1: Detail on Variation of Rent per Acre with Rate of Profits

1.0 Introduction

This post is the start of an attempt to develop an interesting example with both intensive and extensive rent. A feasible technique exists in the example with both intensive and extensive rent. Yet, it is never cost-minimizing. So this example does not do what I want. I have previously thought about other examples.

The example is an extension of my example of the reswitching of the order of rentability. Such reswitching occurs in this example. But the first switch point of the order of rentability is off the frontier.

I think some perturbation of this example will get me an example where a technique with both intensive and extensive rent is cost-minimizing for some range of the rate of profits. That example will illustrate that the orders of efficiency and of rentability can be analyzed in the context of intensive rent. And these orders need not co-incide in the case of intensive rent too.

2.0 Technology, Resources, Final Demand, and Feasibility

Table 1 presents coefficients of production for the example. Two commodities are produced, iron and corn. Aside from the use of land, joint production is not possible. Multiple types of land (that is, three types) exist. Only one agricultural commodity, corn, can be produced on the processes in which land is used. For one type of land, more than one process can be operated on land. Only one process is known for producing iron, the industrial commodity. Each column in Table specifies the person-years of labor, acres of a type of land, tons of iron, and bushels of corn needed to produce a unit output of the specified commodity.

Table 1: The Coefficients of Production

Input	Industry
	Iron	Corn
	I	II	III	IV	V
Labor	1	0.517	91/250	0.67	3/10
Type I Land	0	0.49	0	0	0
Type II Land	0	0	0.59	0	0
Type II Land	0	0	0	9/20	3
Iron	9/20	0.03744	0.0009	0.067	0.08
Corn	2	0.048	0.27	0.15	0.15

I can define various techniques (Table 2) with this technology. I need all twenty-four letters in the Greek alphabet to specify the techniques. Not all techniques are feasible, given technology, endowments, and requirements for use. Land is not scarce for the Alpha, Beta, Gamma, and Delta techniques, and ownership of land obtains no rent. The Epsilon through Upsilon techniques are examples of extensive rent. One type of land obtains a rent in the Epsilon through Xi techniques. All three types are farmed in Omnicro through Upsilon, and two types obtain a rent. Phi is an example of intensive rent. Chi, Psi, and Omega are examples of the combination of intensive and extensive rent.

Table 2: Techniques of Production

Technique	Processes	Land
		Type 1	Type 2	Type 3
Alpha	I, II	Partially farmed	Fallow	Fallow
Beta	I, III	Fallow	Partially farmed	Fallow
Gamma	I, IV	Fallow	Fallow	Partially farmed
Delta	I, V	Fallow	Fallow	Partially farmed
Epsilon	I, II, III	Partially farmed	Fully Farmed	Fallow
Zeta	I, II, IV	Partially farmed	Fallow	Fully Farmed
Eta	I, II, V	Partially farmed	Fallow	Fully Farmed
Theta	I, II, III	Fully Farmed	Partially farmed	Fallow
Iota	I, III, IV	Fallow	Partially farmed	Fully Farmed
Kappa	I, III, V	Fallow	Partially farmed	Fully Farmed
Lambda	I, II, IV	Fully Farmed	Fallow	Partially farmed
Mu	I, III, IV	Fallow	Fully Farmed	Partially farmed
Nu	I, II, V	Fully Farmed	Fallow	Partially farmed
Xi	I, III, V	Fallow	Fully Farmed	Partially farmed
Omnicron	I, II, III, IV	Partially farmed	Fully Farmed	Fully Farmed
Pi	I, II, III, V	Partially farmed	Fully Farmed	Fully Farmed
Rho	I, II, III, IV	Fully Farmed	Partially farmed	Fully Farmed
Sigma	I, II, III, V	Fully Farmed	Partially farmed	Fully Farmed
Tau	I, II, III, IV	Fully Farmed	Fully Farmed	Partially farmed
Upsilon	I, II, III, V	Fully Farmed	Fully Farmed	Partially farmed
Phi	I, IV, V	Fallow	Fallow	Fully Farmed
Chi	I, III, IV, V	Fallow	Fully Farmed	Fully Farmed
Psi	I, II, IV, V	Fully Farmed	Fallow	Fully Farmed
Omega	I, II, III, IV, V	Fully Farmed	Fully Farmed	Fully Farmed

I assume that 100 acres of each of the three types of land are available. Net output consists of 60 tons iron and 80 bushels corn. This completes the specification of the example.

Under these assumptions, Zeta, Lambda, Omnicron, Pi, Sigma, Tau, Upsilon, and Psi are feasible. Types 1 and 3 land are farmed under Zeta, with process IV being operated on Type 3 land. Which of Types 1 and 3 obtain a rent depends on which land is fully farmed. I should say more here.

3.0 Prices of Production

A system of equations specify prices of production for each technique. All operated processes pay the same rate of profits. Rents and wages are paid out of the surplus at the end of the year. A type of land that is only partially farmed is not scarce and pays no rent. I take the net output as the numeraire.

One degree of freedom exists for the system of equations for each technique. Figure 2, below, shows how the wage varies with the rate of profits for each technique. Figure 3 shows the variation in rent per acre with the rate of profits. Figure 1, at the top of this post, is a detail for an interesting part of Figure 3.

Figure 2: Wage Curves for Feasible Techniques

Figure 3: Rent Curves for Feasible Techniques

4.0 Choice of Technique

A technique is not cost-minimizing if it requires a negative rent to be paid. Rent is negative, under Sigma, for both Type 1 and Type 3 lands. Under Zeta, Omnicron, and Pi, rent on Type 3 land is negtive.

This leaves Lambda, Tau, Upsilon, and Psi as feasible techniques that pay positive rents on scarce lands in some range of the rate of profits. Table 1 lists the cost-minimizing techniques, Upsilon and Tau, in order of an increasing rate of profits.

Table 3: Cost-Minimizing Technique

Range	Technique	Order of Efficiency	Order of Rentability
0 ≤ r ≤ 28.49 %	Upsilon	Type 2, 1, 3	Type 2, 1, 3
28.49 ≤ r ≤ 29.05 %			Type 1, 2, 3
29.05 ≤ r ≤ 35.50 %	Tau
35.05 ≤ r ≤ 43.76 %			Type 2, 1, 3

The order of efficiency, at a given rate of profits, is the order in which different types of land would be brought under cultivation as final demand was increased. This order can be read off of Figure 2 by working downward over the wage curves. Since the wage curves for Sigma and for Zeta, Omnicron, and Pi do not intersect, the order of efficiency does not vary, with the rate of profits, in this example. Type 2 land is partially farmed under Sigma. So Type 2 land is first in the order of efficiency. Type 1 land is partially farmed in Zeta, Omnicron, and Pi. Hence, Type 1 land is next in the order of efficiency for techniques in which all three lands are farmed.

The order of rentability is read off of Figures 1 and 3. The order in which rent per acre decreases varies with the rate of profits. For order of rentability differs from the order of efficiency for rates of profits around the switch point between Upsilon and Tau. A change in the order of rentability occurs around the second intersection between the two rent curves for Tau. This effect is a manifestation of the reswitching of the order of rentability. But the order of rentability varies around any intersection of these curves.

It remains to demonstrate that the above claims about which is the cost-minimizing technique at each rate of profits. Figure 4 shows that Lambda is never cost minimizing. Extra profits can always be obtained at Lambda prices by farming Type 2 land with process III. At low rates of profits, extra profits can also be obtained by farming Type 3 land with the other corn-producing process available for that type of land.

Figure 4: Extra Profits at Lambda Prices

Figures 5 and 6 show that Upsilon is cost-minimizing below the switch point, and that Tau is cost-minimizing at higher rates of profits. In these ranges, extra profits are not available by operating the process not in the technique.

Figure 5: Extra Profits at Upsilon Prices

Figure 6: Extra Profits at Tau Prices

Both intensive and extensive rent are paid when Psi is adopted. Figure 7 demonstrates that Psi is never cost-minimizing. Extra profits are always available, whatever the rate of profits.

Figure 7: Extra Profits at Psi Prices

5.0 Conclusion

This post has illustrated the analysis of the choice of technique in an example with both intensive and extensive rent. Constructing the wage curve is not necessarily the correct method of analysis in models with general joint production. Looking at whether or not extra profits are available for the prices associated with a technique is always applicable.