Another 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

I have been exploring examples with both intensive and extensive rent. I try to make this a stand-alone post, with possibilities for later elaborations.

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.

This example works like a lot of mine. It is of perhaps the minimum structure needed to make my points in a model with more than one commodity produced, with a circular structure of production and labor inputs in all processes. Yet its elaboration seems complicated. And surprising results arise here or there. They would be more impressive if they occurred in a larger range of the rate of profits.

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 limitation 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	9/10	3/5	29/50	9/20
Type 1 Land	0	1	0	0	0
Type 2 Land	0	0	49/50	0	0
Type 3 Land	0	0	0	2/5	2
Iron	9/20	1/40	3/2000	29/500	2/30
Corn	2	1/10	9/20	13/100	13/100

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 55 tons iron and 55 bushels corn. This completes the specification of the example. The parameters for the example are fairly arbitrary. They are chosen to ensure reswitching of techniques between the Alpha and Beta techniques when net output is small and no land is scarce. The given net output, however, results in all types of land being scarce.

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 Quantity Flows

What techniques are feasible varies as output expands. I claim that, unlike in models with only extensive rent, the order of efficiency depends on the sequence in which techniques are cost-minimizing as net output expands. So I find the following remark dubious:

"No changes in output and (at any rate in Parts I and Il) no changes in the proportions in which different means of production are used by an industry are considered..." (Sraffa 1960)

Sraffa considers land, along with other examples of joint production, in part II.

If net output is low enough, gross outputs can be such that one type of land is partially farmed. The Alpha, Beta, Gamma, and Delta techniques are all feasible. For the example, the Delta technique is the first technique to become infeasible as output expands. It is replaced by the Eta, Kappa, and Phi techniques, in which type 3 land is scarce and obtains a rent. Table 3 shows which techniques become infeasible or feasible as output expands. Without an improvement in technology, the maximum output for the Omicron, Rho, Tau, and Omega techniques provides a hard limit for this economy.

Table 3: Output Regions

Output Region	Feasible Techniques
1	Alpha, Beta, Gamma, Delta
2	Alpha, Beta, Gamma, Eta, Kappa, Phi
3	Alpha, Gamma, Epsilon, Eta, Kappa, Mu, Xi, Phi
4	Gamma, Epsilon, Eta, Theta, Kappa, Lambda, Mu, Nu, Xi, Phi
5	Gamma, Epsilon, Eta, Theta, Lambda, Mu, Nu, Pi, Phi, Chi
6	Gamma, Epsilon, Theta, Lambda, Mu, Pi, Sigma, Phi, Chi, Psi
7	Gamma, Lambda, Mu, Pi, Sigma, Tau, Upsilon, Phi, Chi, Psi
8	Zeta, Iota, Lambda, Mu, Pi, Sigma, Tau, Upsilon, Chi, Psi
9	Zeta, Iota, Lambda, Mu, Tau, Chi, Psi, Omega
10	Zeta, Lambda, Omicron, Tau, Psi, Omega
11	Omicron, Rho, Tau, Omega

4.0 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

4.1 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.

4.2 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 of zero, 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 wage curve for the Omega technique is only shown for the range in which the rents on all three types of land are zero. The enlargement in Figure 3 shows a range of the rate of profits towards the start of the range at which Omega is cost-minimizing.

Figure 2: Wage Curves for Feasible Techniques

Figure 3: Wage Curves for Feasible Techniques (Detail)

Figure 3 plots rent per acre, as a function of the rate of profits, for the feasible techniques in this post. Figure 1, at the top of this post, is an enlargement of part of the range of the rate of profits. Under Rho, both type 1 and type 3 lands are fully farmed. No range of the rate of profits exist in which scarce land under Rho both receive non-negative rates of profits. Thus, Rho, although feasible, can never be cost-minimizing. Omicron, Omega, and Tau are each uniquely cost-minimizing, other than at switch points, for some range of the rate of profits.

Figure 4: Rent Curves for Feasible Techniques

5.0 Cost-Minimizing Techniques and Orders of Efficiency and Rentability

The cost-minimizing technique, at a given level of net output, varies with the rate of profits. Table 4 specifies the cost-minimizing technique in each output region. The rates of profits shown are only approximate.

Table 4: Cost-Minimizing Technique

Output Region	Range	Technique	Order of Efficiency	Order of Rentability
1	0≤r≤15.8%	Delta	3	N/A
	15.8≤r≤18.1%	Beta	2	N/A
	18.1≤r≤48.2%	Alpha	1	N/A
	48.2≤r≤78.5%	Beta	2	N/A
2	0≤r≤6.2%	Phi	3	3
	6.2≤r≤15.8%	Kappa	3,2	3,2
	15.8≤r≤18.1%	Beta	2	N/A
	18.1≤r≤48.2%	Alpha	1	N/A
	48.2≤r≤78.5%	Beta	2	N/A
3	0≤r≤6.2%	Phi	3	3
	6.2≤r≤15.8%	Kappa	3,2	3,2
	15.8≤r≤16.1%	Xi	2,3	2,3
	16.1≤r≤18.1%	Epsilon	2,1	2,1
	18.1≤r≤48.2%	Alpha	1	N/A
	48.2≤r≤78.5%	Epsilon	2,1	2,1
4	0≤r≤6.2%	Phi	3	3
	6.2≤r≤15.8%	Kappa	3,2	3,2
	15.8≤r≤16.1%	Xi	2,3	2,3
	16.1≤r≤18.1%	Epsilon	2,1	2,1
	18.1≤r≤48.2%	Theta	1,2	1,2
	48.2≤r≤78.5%	Epsilon	2,1	2,1
5, 6	0≤r≤6.2%	Phi	3	3
	6.2≤r≤9.5%	Chi	3,2	3,2
	9.5≤r≤15.4%	Pi	3,2,1	3,2,1
	15.4≤r≤15.8%			2,3,1
	15.8≤r≤16.1%		2,3,1
	16.1≤r≤18.1%	Epsilon	2,1	2,1
	18.1≤r≤48.2%	Theta	1,2	1,2
	48.2≤r≤78.5%	Epsilon	2,1	2,1
7	0≤r≤6.2%	Phi	3	3
	6.2≤r≤9.5%	Chi	3,2	3,2
	9.5≤r≤15.4%	Pi	3,2,1	3,2,1
	15.4≤r≤15.8%			2,3,1
	15.8≤r≤16.1%		2,3,1
	16.1≤r≤16.4%	Upsilon	2,1,3	2,1,3
	16.4≤r≤18.1%			1,2,3
	18.1≤r≤24.4%		1,2,3
	24.4≤r≤48.2%	Tau
	48.2≤r≤50.1%		2,1,3
8	0≤r≤6.2%	Iota	3,2	3,2
	6.2≤r≤9.5%	Chi
	9.5≤r≤15.4%	Pi	3,2,1	3,2,1
	15.4≤r≤15.8%			2,3,1
	15.8≤r≤16.1%		2,3,1
	16.1≤r≤16.4%	Upsilon	2,1,3	2,1,3
	16.4≤r≤18.1%			1,2,3
	18.1≤r≤24.4%		1,2,3
	24.4≤r≤48.2%	Tau
	48.2≤r≤50.1%		2,1,3
9	0≤r≤6.2%	Iota	3,2	3,2
	6.2≤r≤9.5%	Chi
	9.5≤r≤12.2%	Omega	3,2,1	3,2,1
	12.2≤r≤12.5%			3,1,2
	12.5≤r≤12.8%			1,3,2
	12.8≤r≤15.8%			1,2,3
	15.8≤r≤16.1%		2,3,1
	16.1≤r≤18.1%		2,1,3
	18.1≤r≤24.4%		1,2,3
	24.4≤r≤48.2%	Tau
	48.2≤r≤50.1%		2,1,3
10, 11	0≤r≤9.5%	Omicron	3,2,1	3,2,1
	9.5≤r≤12.2%	Omega
	12.2≤r≤12.5%			3,1,2
	12.5≤r≤12.8%			1,3,2
	12.8≤r≤15.8%			1,2,3
	15.8≤r≤16.1%		2,3,1
	16.1≤r≤18.1%		2,1,3
	18.1≤r≤24.4%		1,2,3
	24.4≤r≤48.2%	Tau
	48.2≤r≤50.1%		2,1,3

The range of the rate of profits towards its minimum value illustrates that intensive rent can be transient. When the Delta technique, which produces corn only with process V, becomes infeasible, The technique, Phi in which type 3 land is fully farmed with processes IV and V operating side-by-side, becomes cost-minimizing at a rate of profits of zero and nearby.

For a large enough output, Phi is no longer feasible. The cost-minimizing technique at a rate of profits becomes Iota, a technique with only extensive rent. Type 3 land is fully farmed with process IV, and Type 2 land is partially farmed.

Much else can be said about the sequence in which techniques become cost-minimizing, in various ranges of the rate of profits, as net output expands.

5.1 The Order of Efficiency

The order of efficiency is the order in which techniques are adopted with increasing net output at a given wage or rate of profits. The order of efficiency is also known as the order of fertility. Since the order of efficiency varies with the rate of profits, fertility is not defined solely by physical inputs and outputs.

In models with extensive rent, the order of efficiency can be read off the order of wage curves. The range of the rate of profits at which Tau is cost-minimizing in the example illustrates. The order of efficiency here is the sequence of wage curves from the highest to the wage curve for Gamma. Under Gamma, Type 3 land is partially farmed with process IV, as in the Tau technique.

The order of efficiency for Tau varies at the switch point between the Alpha and Beta wage curves. For the lowest levels of net output, this switch point between Alpha and Beta is an example of capital-reversing. But, for Tau, a change in quantity flows is not even associated with this change in the order of efficiency. Capital-reversing does not occur.

Consider the range of the rate of profits between approximately 6.18 and 9.54 percent. As net output expands, the Delta, Kappa, Chi, and Omicron techniques succeed one another. Delta and Kappa operate process V on type 3 land. Chi operates a linear combination of processes IV and V. The sequence of wage curves in this range varies from Delta, Gamma, Beta, Alpha through Delta, Beta, Gamma, Alpha. Yet the order of efficiency does not vary. Even though process IV is operated on type 3 land in both Gamma and Omicron, the switch point between Beta and Gamma has no effect on the order of fertility. Gamma does not matter to the initial historical order in which techniques are introduced here.

For Omega, the switch point between the Alpha and Gamma techniques does not matter for the order of efficiency. More could be set about the variation in the order of efficiency with the rate of profits.

5.2 The Order of Rentability

The order of rentability varies with intersections of rent curves. These variations in the order of rentability are independent from variations in the order of efficiency. The owner of a less fertile land can obtain more rent per acre than the owner of a more fertile land. As usual, a simplistic marginal productivity theory of distribution cannot be sustained.

5.3 The Cost-Minimizing Technique for the Example

Observations have been made above about the cost-miniziming technique. But the interaction between quantity flows, prices, and rents suggest that the cost-minimizing technique cannot always be found by constructing a frontier from the wage curves.

Figure 5 justifies that Omicron is cost-minimizing at a small rate of profits. Capitalists can gain extra profits by adopting process V in the range of the rate of profits at which both Omicron and Omega pay positive rents. Type 5 land becomes fully farmed by combining the two processes on Type 3 land, and the Omega technique results.

Figure 5: Extra Profits at Omicron Prices

Figure 6 justifies that Tau is cost-minizing at a high rate of profits. Here, too, the Omega technique will be adopted in the range of the rate of profits at which both the technique under consideration and Omega pay positive rents.

Figure 6: Extra Profits at Tau Prices

6.0 Conclusion

This post has illustrated that the introduction of natural resources into a model of the production of commodities provides a complicated picture. The concepts of the order of efficiency and the order of rentability apply to a model in which both intensive and extensive rent occur, depending on net output and the range of the rate of profits. The order of efficiency depends on the sequence in which techniques are cost-minimizing as net output expands, in a more complicated way than when just extensive rents are available. Intensive rent can be transient. An expansion of net output may replace a technique in which intensive rent obtains, at a given rate of profits, with a technique with only extensive rent. And the opposite can happen, as well.

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.

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.

Extensive Rent And Labor Values

1.0 Introduction

Do scarce natural resources provide additional difficultes for modern reconstructions of classical and Marxian theories of value? After all land can be sold or rented, and labor cannot produce more land. (I put aside Holland.)

This post presents an exposition of the theory of extensive rent, a start on examining possible difficulties. This type of rent provides the least dificulties, as I understand it, for such modern reconstructions. As usual, I present an example, close to the minimal complexity, needed to make my points. The model can obviously be generalized to include many more produced industrial commodities; many more types of agricultural commodities; and many more types of land, each specialized to support the production of one kind of agricultural commodity.

2.0 Technology

Table 1 specifies the technology for this example. Each column defines the coefficients of production for a process. For example, the only iron-producing process requires a_0,1 person-years of labor, a_1,1 tons of iron, and a_2,1 bushels of corn as inputs for every ton iron produced. I assume that each process requires a year to complete and exhibits constant returns to scale. The corn-producing processes each have an upper limit on how much corn they can produce.

Table 1: A Technology

	Iron Industry	Corn Industry
	Process a	Process b	Process c
Labor	a0,1	a0,2	a0,3
Land, Type 1	c1,1 = 0	c1,2 > 0	c1,3 = 0
Land, Type 2	c2,1 = 0	c2,2 = 0	c2,3 > 0
Iron	a1,1	a1,2	a1,3
Corn	a2,1	a2,2	a2,3
OUTPUTS	1 ton iron	1 bushel corn	1 bushel corn

I assume two types of land exist, distinguished by the processes that can be operated on them. A single corn-producing process can be operated on each type of land. Only a certain number of acres of each type of land exists. Each corn-producing process leaves the land unchanged at the end of operating the process. The given quantities of land limit how much corn can be produced. This model cannot accomodate a positive steady-state rate of growth without technical progress.

A full specification for this model should include requirements for use. I assume that the net output must be such that both types of land are farmed, but only one type is fully farmed. Two techniques for production exist, as shown in Table 2. All three processes are operated in each technique, but only one type of land is fully used.

Table 2: Specification of Techniques

Technique	Type 1 Land	Type 2 Land
Alpha	Partially farmed	Fully farmed
Beta	Fully farmed	Partially farmed

3.0 Parameters and Variables

I have already implicitly defined certain parameters above. Table 3 lists certain parameters I use in this model. Table 4 lists variables that I need. Some assumptions are imposed on the matrices A_α and A_β:

• All produced commodities are basic. Iron and corn enter directly or indirectly into the production of both commodities.
• The technology expressed by these matrices is productive. Each matrix satisfies the Hawkins-Simon condition.

Table 3: Selected Parameters

Symbol	Definition
a0, α	Two-element row vector consisting of first two labor coefficients.
a0, β	Two-element row vector consisting of first and third labor coefficients.
Aα	2x2 matrix, with columns consisting of iron and corn coefficients of production for first and second processes.
Aβ	2x2 matrix, with columns consisting of iron and corn coefficients of production for first and third processes.
d	Two-element column vector consisting of iron and corn quantities in the numeraire.

Table 4: Variables

Symbol	Definition
vα	2-element row vector of labor values when type 1 land is free.
vβ	2-element row vector of labor values when type 2 land is free.
p	2-element row vector of prices of unit quantities of iron and corn.
p1	The price of iron, in numeraire units per ton. The first element of p.
p2	The price of corn, in numeraire units per bushel. The second element of p.
rho1	The rent of type 1 land, in numeraire units per acre.
rho2	The rent of type 2 land, in numeraire units per acre.
w	The wage, in numeraire units per person-year.
r	The rate of profits.

4.0 Labor Values

Given the technique in use, how much additional labor would be employed throughout the economy if the net output was such that one additional unit of iron were produced? This is the labor value of iron, and it easily calculated in the theory. The answer to the same question for corn is its labor value.

Suppose type 1 land is free. Then labor values are:

v_α = a_{0, α} (I - A_α)^-1

Labor values, when type 2 land is free, are the corresponding Leontief employment multipliers for the Beta technique. Variations in net output require varying the amount of the land farmed on the type of land that is not fully farmed.

5.0 Prices of Production

With market prices, some operated processes will be obtaining a higher rate of profits than average, and some will be obtaining a lower rate. These variations in the profit rates are perhaps a signal to capitalists that they should disinvest in some industries or processes and increase investment in others. Models of cross-dual dynamics and other models explore these disequilibria.

Prices of production are such that these signals are absent. All operated processes obtain the same rate of profits. I assume profits, rents, and wages are paid out of the surplus product at the end of the year. The following three equations express the condition that all processes obtain the same rate of profits:

(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

The next equation expresses the condition that the price of the numeraire is unity:

p d = 1

Finally, one of the rents must be zero:

rho₁ rho₂ = 0

The last equation is a defining feature of the theory of extensive rent.

Suppose one of the types of land is rent-free. For deiniteness, let type 1 land be only partially farmed. Then the first four equations are in terms of five variables (p₁, p₂, rho₂, w, r). Just as in the case with only circulating capital, prices of production are specified up to one degree of freedom. In classical political economy, the wage is take as given.

6.0 Choice of Technique

Suppose the wage is non-negative and does not exceed a maximum defined by the technology. The system of equations for prices of production has two solutions. Each solution has the rent on one type of land set to zero. The cost-minimizing technique is the one in which the rent on the other land is positive. If, for a technique, the rent on a type of land is negative, that technique will not be adopted by capitalists. At a switch point, the rents on both types of land are zero.

But the analysis of the choice of technique can be expressed in terms of wage curves. Suppose rents were zero. Consider the first two equations in the system of equations for the prices of production and the equation setting the price of the numeraire to unity. These equations yield a function in which the wage decreases with an increase in the rate of profits. Similarly, the first and third equations yield another decreasing wage curve.

In the case of circulating capital alone, the cost-minimizing technique is found by the wage frontier formed out of the outer envelope of these wage curves. At a given wage, the cost-minimizing technique maximizes the wage.

In this example of extensive rent, the cost-minimizing technique is found by the wage frontier formed out of the inner envelope of the wage curves.

In either case, the appropriate wage frontier shows that a lower rate of profits is associated with a higher wage and vice versa. The maximum wage occurs when the rate of profits is zero. The maximum rate of profits arises when the wage is zero.

7.0 Special Cases

Which land is free and which land pays a rent depends on either the wage or the rate of profits, whichever is taken as exogenous in the system of prices of production. At any rate, a wage frontier exists in which the wage is higher the smaller the rate of profits. This frontier is not the outer frontier of the wage curves for the technique.

Without loss of generality, suppose the Alpha technique is cost-minimizing. Type 1 land is not fully farmed and pays no rent. Then labor values are defined, based on the iron-producing process and the process on type 1 land.

Consider the special case in which a_{0, α} is an eigenvector corresponding to the maximum eigenvector for A_α. Then relative prices of production are equal to relative labor values.

On the other hand, suppose that the numeraire is the standard commodity, as found from a_{0, α} and A_α. Suppose only the standard commodity is produced. In this case, only the process on the rent-free land would be used, in contradiction to the analysis of the choice of technique. And suppose the wage is paid out in the form of the standard commodity. Then the following hold:

• The labor value of gross output is equal to total gross output, evaluated at prices of production.
• The labor value of net output is equal to net output, evaluated at prices of production.
• The labor value of the proportion of the standard commodity paid out in wages is equal to wage goods, evaluated at prices of production.

This special case seems especially forced in the case of extensive rent. Is some reformulation available in which surplus value can be treated as the sum of profits and rent?

I do not address the use of labor values in Marx's account of exploitation, Marx-biased technical change, and so on. The special cases in which the labor theory of value hold make obvious that, for a given technology, a higher rate of profits require a lower wage. And this wage frontier continues to hold in models of extensive rent.

8.0 Conclusion

The inclusion of natural resources, insofar as they can be modeled by extensive rent, does not seem to pose any additional issues for modern formulations of classical and Marxian political economy. It does highlight some issues that arise in models with circulating capital.

Labor values can be calculated for all produced commodities, given the technique in use. They are calculated from the marginal land that receives no rent. But suppose that a choice of technique exists. Then, an analysis at the level of prices of production must be prior to the calculation of labor values. The theory of extensive rent highlights this issue.

As Ricardo and Marx noted, prices of production are generally not proportional to labor values. They are equal in the special case, in which all industries have equal organic compositions of capital, in both models of circulating capital and of extensive rent. In the latter case, the organic composition of capital is found for agriculture from no-rent lands partially farmed.

A commodity of average organic composition is picked out in both models. Total labor values and the labor value of wages are equal to the corresponding aggregates in the system of prices of production when this average commodity is used as numeraire and is produced. These invariants, though, have to restricted to the production of the numeraire with the iron-producing process and the process on no-rent land. It is not clear to me that Marx thought his invariants held in his chapters on rent, given their location towards the end of volume 3 of Capital.

Obviously, these observations on natural resources and rent are just a start. They do seem to match what Ricardo was about in the second chapter of his Principles. The analysis of the choice of technique can be thought of, somewhat, as a critique of Ricardo.

At any rate, prices of production are well-defined in models of extensive rent. And they can be used in an analysis of the choice of technique. As usual, I present the analysis with no mention of utility maximization, preferences, or tastes.