Intensive Rent With Two Types Of Land

Figure 1: Wages Curves for Example of Intensive Rent

1.0 Introduction

This post modifies an example from Antonio D'Agata. Two types of land exist, each specialized for producing a specific commodity.

In the example, some wage curves slope upwards, which is not possible in a model with circulating capital alone. The cost-minimizing technique is not found from the outer frontier of the wage curves. For one range of the rate of profits, no cost-minizing technique exists, even though a feasible technique exists in that range with a positive wage and positive prices. If the wage is taken as given, more than one cost-minizing technique exists in the range of wages where a cost-minimizing technique exists.

This example does not illustrate variation in the order of rentability with the wage or rate of profits. Hence it also does not illustrate the reswitching of the order of rentability.

2.0 Technology, Endowments, Requirements for Use

Table 1 provides the technology for such an example. Each column specifies the quantities of labor, iron, wheat, and rye needed to produce a unit output of the commodity produced by the corresponding industry. The table also specifies the quantity of land that must be rented to operate that process. Constant returns to scale are assumed, with the limitation that the endowments of each type of land are givens.

Table 1: The Coefficients of Production

Input	Industry
	Iron	Wheat		Rye
	I	II	III	IV	V
Labor	a0,1 = 1	a0,2 = 1	a0,3 = 2	a0,4 = 1	a0,5 = 1
Type 1 Land	0	c1,2 = 1	c1,3 = 1	0	0
Type 2 Land	0	0	0	c2,4 = 1	c2,5 = 1
Iron	a1,1 = 0	a1,2 = 0	a1,3 = 1/100	a1,4 = 1/10	a1,5 = 1/10
Wheat	a2,1 = 0	a2,2 = 0	a2,3 = 0	a2,4 = 2/5	a2,5 = 1/10
Rye	a3,1 = 1/10	a3,2 = 3/5	a3,3 = 11/20	a3,4 = 1/10	a3,5 = 2/5

I show each type of land as specialized to produce a different kind of agricultural commodity. The givens include the amount of each type of land available. Let t₁ be the acres of type 1 land available and t₂ be the acres of type 2 land available:

• t₁: 97 acres.
• t₂: 100 acres.

The column vector d representing the requirements for use has components:

• d₁: 90 tons iron.
• d₂: 60 quarters wheat.
• d₃: 19 bushels rye.

This vector d of net ouput is also the numeraire.

Table 2 specifies the techniques. All three commodities are Sraffa basics in all techniques. Only the Gamma, Zeta, Eta, and Iota techniques are feasible. For a technique to be feasible, the processes comprising the technique can be operated at a level to produce the required net output.

Table 2: Is Process Operated By Technique?

Input	Industry
	I	II	III	IV	V
Alpha	Yes	Yes	No	Yes	No
Beta	Yes	Yes	No	No	Yes
Gamma	Yes	No	Yes	Yes	No
Delta	Yes	No	Yes	No	Yes
Epsilon	Yes	Yes	No	Yes	Yes
Zeta	Yes	No	Yes	Yes	Yes
Eta	Yes	Yes	Yes	Yes	No
Theta	Yes	Yes	Yes	No	Yes
Iota	Yes	Yes	Yes	Yes	Yes

3.0 Prices of Production

The equations for prices of production vary among the techniques. Both lands are in excess supply and pay no rent for Gamma. Type 2 land is fully farmed under Zeta, and it pays a rent. Type 1 land pays a rent under Eta. Both lands pay a rent under Iota.

In the usual notation, the equations for prices of production for Iota are:

(p₁ a_1,1 + p₂ a_2,1 + p₃ a_3,1)(1 + r) + w a_0,1 = p₁

(p₁ a_1,2 + p₂ a_2,2 + p₃ a_3,2)(1 + r) + rho₁ c_1,2 + w a_0,2 = p₂

(p₁ a_1,3 + p₂ a_2,3 + p₃ a_3,3)(1 + r) + rho₁ c_1,3 + w a_0,3 = p₂

(p₁ a_1,4 + p₂ a_2,4 + p₃ a_3,4)(1 + r) + rho₂ c_2,4 + w a_0,4 = p₃

(p₁ a_1,5 + p₂ a_2,5 + p₃ a_3,5)(1 + r) + rho₂ c_2,5 + w a_0,5 = p₃

The price of the numeraire is unity:

p₁ d₁ + p₂ d₂ + p₃ d₃ = 1

These equations can be solved, given either the wage or the rate of profits. Figure 1, at the top of this post, shows the resulting wage curves for the feasible techniques. Figure 2 shows the rent curves.

Figure 2: Rent Curves for the Example

4.0 The Choice of Technique

For a low rate of profits, Iota is the cost-minimizing technique. For a rate of profits greater than that at the switch point for the Iota and Zeta wage curves, the Zeta technique is cost-minimizing, up to the maximum rate for Zeta. No technique is cost-minimizing for a larger rate of profits. Type 2 land has a larger rent per acre than type 1 land in the full range of the rate of profits in which a cost-minimizing technique exists. Type 1 land obtains a rent in the range in which Iota is cost-minimizing. It is free when Zeta is cost-minimizing.

These conclusions are justified by looking at which processes can obtain extra processes when prices of production for a given technique rule. Figure 3 graphs extra profits as a function of the rate of profits with prices of production for the Gamma technique. Either both the third and fifth process can obtain extra profits or the fifth process alone, depending on the level of the rate of profits. Thus, the Gamma technique is never cost-minimizing.

Figure 3: Extra Profits at Gamma Prices

Suppose the rate of profits is given and is in the range from the maximum rate of profits for the Zeta technique to the maximum rate of profits for the Gamma technique. Gamma is feasible, but not cost-minimizing. At prices of production for Gamma, managers of firms would want to operate the fifth process, which produces rye. Firms would adopt either the Delta or the Zeta technique, depending on whether they continue to operate the fourth process at some level. But Delta is infeasible, and prices of production for Zeta are such that the wage is negative in this range. Thus, no cost-minizing technique exists here.

Figure 4 shows extra profits for prices of production for Zeta. At a rate of profits less than the rate at the the switch point for Zeta and Iota, operating the second process obtains extra profits. Iota can be adopted in this range. Zeta is cost-minimizing for a higher rate of profits, up to the maximum rate of profits for Zeta.

Figure 4: Extra Profits at Zeta Prices

Figure 5 graphs extra profits for prices of production for the Eta process. Eta is never cost-minimizing. Operating the fifth process at Eta prices obtains extra profits.

Figure 5: Extra Profits at Eta Prices

Prices of production for Iota are such that extra profits are not obtained in operating any of the five processes in the technology. As long as the wage and rent on both types of land are non-negative, at a given rate of profits, Iota is cost-minimizing.

5.0 Conclusion

Type 2 land obtains a rent for the cost-minizing technique in the full range of the rate of profits where a cost-minimizing technique exists. Type 1 land only obtains a rent for a low range of the rate of profits.

Certain properties of models of circulating capital do not generalize, annoyingly, to the theory of joint production. Are there any such properties that are violated in models of pure joint production that are not also violated in models of intensive rent?