An Example With A Cost-Minimizing Technique With Intensive And Extensive Rent

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

1.0 Introduction

Consider a model of the production of commodities with non-produced means of production that are unchanged by their use in production. In other words, they are types of land. In a simple model of extensive rent, a single agricultural commodity, 'corn', can be produced, on each type of land, with a single production production. This post expands a simple multi-commodity model to postulate the existence of two production processes on one type of land. The model then combines intensive and extensive rent, depending on the choice of technique.

In the example, all three types of land are at least partially cultivated to satisfy requirements for use. Whether or not all three types of land obtain a rent depends on the level of profits. A mixture of intensive and extensive rent is obtained only for a range of the rate of profits.

I repeat a lot from a previous post so that this post somewhat makes sense by itself.

2.0 Technology, Resources, Final Demand, and Feasibility

A model of the production of commodities is specified by the technology, the endowments of unproduced natural resources, and the requirements for use. Technology is specified, in a discrete technology, by coefficients of production for each production process. Each process is assumed to require the same time to complete and to exhibit constant returns to scale, up to the limited imposed by scarce land. The endowments of each type of land are specifed. Requirements for use are specified by final demand.

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 1 Land	0	0.49	0	0	0
Type 2 Land	0	0	0.59	0	0
Type 3 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

Various techniques (Table 2) can be defined with this technology. All twenty-four letters in the Greek alphabet are needed 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 66 tons iron and 88 bushels corn. This completes the specification of the example. The parameters for the example are fairly arbitrary. They are chosen to ensure a reswitching of the order of rentability for the Tau technique and to ensure that the Omega technique is feasible.

Under these assumptions, Omnicron, Rho, Tau, and Omega are feasible. All three types of land are farmed under these three techniques. Type 1 land is only partially farmed under Omnicron, and it is non-scarce and does not obtain a rent. Type 2 land does not obtain a rent under Rho. Type 3 land does not obtain a rent under Tau. All three types are fully farmed under Omega. A linear combination of processesare IV and V are operated side-by-side under Omega. Type 3 land is therefore scarce under Omega. All three types are farmed under Omnicron, with non-scarce Type 3 land only partially farmed.

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.

As an example, the system of equations in following five displays specify the prices of production for the Omega technique.

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

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

(p₁ a_1,3 + p₂ a_2,3)(1 + r) + rho₂ c_2,3 + w a_0,3 = p₂

(p₁ a_1,4 + p₂ a_2,4)(1 + r) + rho₃ c_3,4 + w a_0,4 = p₂

(p₁ a_1,5 + p₂ a_2,5)(1 + r) + rho₃ c_3,5 + w a_0,5 = p₂

Prices of production for the other techniques are specified by a subset of the system of equations for the Omega technique. Each operated process corresponds to an equation in the corresponding system of prices of production. The rent on land that is partially farmed is zero in the corresponding equation, since land in excess supply is not scarce.

The numeraire is specified by a further equation, where the column vector d represents net output.

p₁ d₁ + p₂ d₂ = 1

3.1 On the Solution

A linear combination of the last two equations in the system of prices of production, for the Phi, Chi, Psi, and Omega techniques, eliminates the rent of type 3 land. In the techniques with extensive rent, one of the equations for a corn-producing process does not contain a term for rent either.

This equation for a corn-producing process or the linear combination of the last two equations can be combined with the first equation, for the iron-producing process. This results in a system of two equations in four unknowns, the price of iron, the price of corn, the wage, and the rate of profits. The equation for the numeriare removes one degree of freedom. If the rate of profits is taken as given, this is a linear system which can be solved for prices of produced commodities and the wage.

The rent per acre can be found for each equation remaining in the original system of equations for a technique. The Alpha, Epsilon, Zeta, Eta, Omnicron, and Pi techniques, for example, have the same solution for prices of produced commodities and the wage. Epsilon, Omnicron, Pi have the same rent per acre on type 2 land. Zeta and Omnicron have the same rent per acre on Type 3 land, while Eta and Pi have the same rent per acre on Type 3 land.

3.2 Wage and Rent Curves

Given the technique, the wage is therefore a function of the rate of profits. Likewise the rent on lands that are always fully-farmed with that technique is also a function of the rate of profits.

The wage is a declining function of the rate of profits in the first four techniques and in the 16 techniques with extensive rent alone. A maximum wage corresponds to a rate of profits, and a maximum rate of profits corresponds to a wage of zero. The wage curve can be upward-sloping in models of extensive rent. The wage curves, in the example, happen to be downward-sloping in the example. Figure 2 shows the wage curves for the feasible techniques in the example. The order of efficiency is the order in which techniques are adopted with increasing net output at a given wage or rate of profits. In models with extensive rent, the order of efficiency can be read off the order of wage curves.

Figure 2: Wage Curves for Feasible Techniques

Figure 3 shows the rent curves for the techniques with non-negative rents in the example. Figure 1, at the top of the post, is an enlargement. Rent curves do not need to have any particular slope. They can slope down or up and vary along their extent. The rent curves for Tau are an example of the reswitching of the order of rentability.

Figure 3: Rent Curves for Feasible Techniques

4.0 The Choice of Technique

Only two techniques, Tau and Omega, are feasible in the example and have non-negative rents for scarce lands. Table 3 lists approximate ranges of the rate of profits and which techniques are cost-minimizing in which ranges. The orders of efficiency and the order of rentability are also shown.

Table 3: Cost-Minimizing Technique

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

Figure 4 justifies which technique is cost-minimizing in which range of the rate of profits. Capitalists can gain extra profits by adopting process V in the range in which Omega pays positive rents. Type 5 land becomes fully farmed by combining the two processes on Type 3 land, and the Omega technique results. For higher rates of profits, Tau is cost-minimizing, up to the maximum for Tau.

Figure 4: Extra Profits at Tau Prices

Intensive and extensive rents are both obtained by landlords when the Omega technique is cost-minimizing. Whenever the Omega technique is cost-minimizing, and in some range of the rate of profits in which Tau is cost-minimizing, the order of efficiency varies from the order of rentability. Type 2 land is more efficienct or more fertile than Type 1 land. Yet ownership of Type 1 land obtains more rent per acre than Type 2 land. Why would one ever expect competitive capitalist markets to reward efficiency?

5.0 Conclusion

This post presents the first concrete example of a case where a cost-minimizing technique combines intensive and extensive rent. It demonstrates that the concepts of the order of effiency and the order of rentability apply to models with intensive rent. As with models with only extensive rent, the order of effiency cannot be generally defined in terms of physical properties alone. And these orders can differ from one another at some given wage or rate of profits.

The example does not illustrate issues that can arise with intensive rent. Wage curves can slope up. The cost-minimizing technique can be non-unique away from switch points. No cost-minimizing technique may exist, even though feasible techniques exist at a given wage or rate of profits (D'Agata 1983).

The analysis can be extended to more kinds of rent and more complicated production models, while still not treating general joint production. Absolute rent, which may not make sense (Basu 2022) and external intensive rent (Kurz and Salvadori 1995) are examples. Rent might be analyzed in models with systematic, persistent variations in the rate of profits among industries (Vienneau 2024). Likewise, a more general model could have some types of lands that are inputs into processes that each produce a different agricultural commodity. Does it make sense to compare and contrast the order of efficiency and the order of rentability in these models?

References

• D’Agata, A. 1983. The existence and unicity of cost-minimizing systems in intensive rent theory, Metroeconomica 35: 147-158.
• Basu, Deepankar. 2022. A reformulated version of Marx's theory of ground rent shows that there cannot be any absolute rent. Review of Radical Political Economics 54(4): .
• Kurz, H. D. and Salvadori N. 1995. Theory of Production: A Long-Period Analysis, Cambridge: Cambridge University Press.
• Quadrio-Curzio, A. 1980. Rent, income distribution, and orders of efficiency and rentability, in Pasinetti, L. L. (ed.) Essays on the Theory of Joint Production, New York: Columbia University Press.
• Quadrio-Curzio, A. and F. Pellizzari. 2010. Rent, Resources, Technologies. Berlin: Springer. [I NEED TO READ THIS TO ENSURE THAT I AM ORIGINAL]
• Vienneau, R. L. 2022. Reswitching in a model of extensive rent. Bulletin of Political Economy 16(2): 133-146.
• Vienneau, R. L. 2024. Characteristics of labor markets varying with perturbations of relative markups. Review of Political Economy (36)2: 827-843.