Existence Of A Cost-Minimizing Solution In A Model With Extensive And Intensive Rent

1.0 Introduction

This post presents a special case model combining extensive and intensive rent. No joint production, other than that associated with land, exists in the model. Only one agricultural commodity, 'corn', is produced. Each corn-producing process operates on one type of land. No possibility exists of simultaneously using two or more unproduced natural resources.

But the more restrictive conditions are on land coefficients. The processes that operate on each type of land can be strictly ordered by the acres per bushel corn produced. Ties do not exist. Furthermore, the coefficients of production are such that no negative values arise when taking a linear combination of two processes to eliminate land. These assumptions rule out, for example, certain non-existence and non-uniqueness examples from D'Agata (1983). I do not claim that they are justified by economic reasoning. This post is an exploration of the boundary between models that share properties of models of circulating capital and models with joint production that do not have those properties.

As far as I know, this special case model, in which problems of general joint production do not arise, is novel. It fills a gap in Kurz & Salvadori (1995). Bidard and Erreygers have a series of papers developing the theory of rent. They apply the Lemke algorithm. The Lemke algorithm informs the user if a solution does not exist. Thus, they have no need to state the special case assumptions that I do.

I also do not know that anybody has noted the possibility of the orders of efficiency and rentability being entirely opposite in some range of the rate of profits. I have not adequately emphasized this demonstration in previous expositions of my numerical example.

This work requires a proof of the existence theorem to be complete.

2.0 Parameters and Variables

A model combining extensive and intensive rent is developed here. Tables 1 and 2 specify notation for the parameters and variables of the model.

Table 1: Parameters

Symbol	Definition
n	Number of produced commodities. Positive.
m	Number of processes in the technology, with m ≥ n.
k	Number of types of land available. Positive.
a0	A m-element row vector. Each element is the person-years needed to operate a process at a unit level. All elements are positive.
A	A n x m input matrix. Each column specifies the physical inputs of produced commodities needed to operate a process at unit level.
B	A n x m output matrix. Each column is the physical outputs from operating a process at unit level.
C	A k x m input matrix for land. ci,j is the acres of the ith type of land needed as input when the jth process is operated at unit level.
t	A k-element column representing endowments. Each element is the number of acres of a type of land available. All elements are positive.
d	A n-element column representing requirements for use and the numeraire. Each element is the physical quantity of a commodity that must be in net output.

Table 2: Variables

Symbol	Definition
q	A m-element column vector. The elements of the vector are the levels at which the processes are operated.
p	A n-element row vector of prices.
rho	A k-element row vector of rents.
r	The rate of profits
w	The wage, in numeraire units per person-year.

3.0 Assumptions and the Structure of Input and Output Matrices

I start out with some abstract assumptions:

• All input and output coefficients are non-negative.
• Direct labor is needed to operate each process. All elements of a₀ are positive
• Commodity inputs are needed for each process. Each column of A has some positive entries.
• No pure joint production, other than land, is possible. Each column of B contains exactly one positive entry. In fact, that entry is unity.
• Some process produces each (non-land) commodity. Each row of B contains at least one positive entry.

The input and output matrices have a specific structure. The produced commodities consist of n - 1 industrial commodities and one agricultural commodity, corn. More specifically, the output matrix has the structure in Figure 1. The subscripts represent the size of each submatrix. The upper left submatrix is the identity matrix. The upper right is a matrix of all zeros. The lower left is a row vector of zeros. And the lower right submatrix is a unit row vector. The first n - 1 processes produce the industrial commodities. The remaining processes produce corn.

Figure 1: Structure of Output Matrix

The input matrix for land is assumed to have a certain structure too (Figure 2). Land is not needed as a direct input to produce the industrial commodities. The elements of the first n - 1 columns of C are all zero. Each process for producing corn requires an input of the services of one type of land. That is, each of the last m - n + 1 columns of C contain exactly one non-zero element. Each type of land is used in at least one process for producing corn. Each row of C contains at least one non-zero element.

Figure 2: Structure of Land Input Matrix

With these assumptions, the kind of rent that can be obtained by landlords depends on the number of produced commodities, the number of production processes in the available technology, and the number of types of land:

• If k = m - n + 1, the coefficients of production specify a model of extensive rent alone.
• If k < m - n + 1, the parameters specify a model with intensive rent.
• If k = 1 and n < m, this is a model of intensive rent alone.

Models with extensive rent alone or with intensive rent alone are thus special cases of this model.

4.0 Assumptions on Solving Subsystems

Each technique is associated with a solving subsystem (Quadrio Curzio & Pellizzari 2010), as defined by a n-element row vector â₀^h and a n x n matrix Â^h. A solving subsystem resembles the vector of direct labor coefficients and the input-output matrix for a model with circulating capital alone. The first (n - 1) labor coefficients and columns in the solving subsystems are from the industrial processes specified by the technology. The last labor coefficient and last column are from a corn-producing process or a linear combination of a pair of corn-producing processes.

Only the first and last corn-producing processes on a type of land have a technique with a solving subsystem for extensive rent. With the structure of the land input matrix, the first solving subsystem with extensive rent is as in Figure 3. The second solving subsystem with extensive rent is as in Figure 4.

Figure 3: First Solving Subsystem with Extensive Rent

Figure 4: Second Solving Subsystem with Extensive Rent

Solving subsystems for successive pairs of processes on a type of land are techniques with intensive rent. The price equation for the first process on the first type of land is given by:

p a_.,n (1 + r) + rho₁ c_{1, n} + w a_0,n = p_n

The price equation for the second process is given by:

p a_{.,n + 1} (1 + r) + rho₁ c_{1, n + 1} + w a_{0,n + 1} = p_n

A linear combination of these equations can eliminate rent:

p [(c_{1,n + 1} a_.,n - c_1,n a_{.,n + 1})/(c_{1,n + 1} - c_1,n)] (1 + r)

+ w [(c_{1,n + 1} a_0,n - c_1,n a_{0,n + 1})/(c_{1,n + 1} - c_1,n)] = p_n

This 'process' provides the coefficients for the last column in the first solving subsystem for intensive rent.

For a linear combination to have non-negative levels of operation of the original two processes, the level of operation q' of the corn-producing process in the solving subsystem must satisfy a condition like the following:

t₁/c_{1,n + 1} ≤ q' ≤ t₁/c_1,n

A type of land that has only one corn-producing process available to operate on it has no solving subsystems for intensive rent. It has one solving subsystem, for extensive rent.

The matrices in the solving subsystems are assumed to meet the following conditions.

• All commodities are basic in each technique with a solving subsystem. Each commodity enters directly or indirectly into the production of all commodities. The input matrices in the solving subsystems are indecomposable.
• All input matrices for a solving subsystem are productive. Each matrix satisfies the Hawkins-Simon conditions. A level of operations of the processes exists such that a positive net output exists.
• All direct labor coefficients and input coefficients are non-negative. For example, for the first solving subsystem with intensive rent:

c_1,n a_{l,n + 1} ≤ c_{1,n + 1} a_l,n, l = 0, 1, ..., n

The last assumption seems to have little economic meeting. But it restricts this model to one that closely resembles the circulating capital case.

5. 0 The Model

The model of extensive and intensive rent is specified in terms of certain equalities and inequalities.

I start with quantity flows. Levels of operation satisfy requirements for use:

(B - A) q = d

Endowments of land are not exceeded:

C q ≤ t

A vector is less than or equal to another if and only if all elements of the first vector are less than or equal to the elements of the second. All levels of operation are non-negative:

q ≥ 0

The equality and two inequalities specify the quantity system.

No pure economic profits are available in any process:

p A (1 + r) + rho C + w a₀ ≥ p B

All prices are non-negative:

p ≥ 0

All rents are non-negative:

rho ≥ 0

The above three inequalities specify the price system.

The rule of free goods states, in this context, that lands in excess supply pay no rent

rho [C q - t] = 0

The rule of non-operated processes states that processes in which costs exceed revenues are not operated:

[p B - p A (1 + r) - rho C - w a₀] q = 0

The rule of free goods and the rule of non-operated processes are duality conditions.

A solution, given the rate of profits, is a vector of levels of operation of each process, a wage, a price of each produced commodity, a rent for each type of land that satisfices the price system, the quantity system, and the duality conditions.

6.0 Existence Theorem

Theorem: Let the rate of profits be given and less than the maximum. Suppose the vector of direct labor coefficients, the land input matrix, the input matrix, the output matrix, and the solving subsystems satisfy the conditions in Sections 3 and 4. Furthermore, suppose the given net output d can be feasibly produced with the technology. Then a (cost-minimizing) solution to the model of extensive and intensive rent in Section 5 exists.

7.0 Conclusion

I claim that, whatever the economic sense of these assumptions, they imply that wage curves for the price systems associated with each solving subsystem are decreasing. Each solving subsystem has a maximum rate of profits, at a wage of zero, and a maximum wage, at a rate of profits of zero. In other words, they resemble the wage curves in single-product (circulating capital) models. D'Agata's examples show that a more general model does not have these properties. I think Guido Erreygers has some interesting examples, too.

The proof proceedes by working downward, at a given rate of profits, from the highest wage to the lowest, in the wage curves for the solving systems. A subtle point is that not all wage curves can be for a cost-minimizing solution, aside from feasibility, I think. I would like a criterion for removing wage curves from the ordered list of wage curves prior to working through the solving subsystems.

Selected References

• Antonio D'Agata. 1983. The existence and unicity of cost-minimizing systems in intensive rent theory. Metroeconomica 35 (1-2): 147-158.
• Christian Bidard. 2010. The dynamics of intensive cultivation. Cambridge Journal of Economics 34: 1097-1104.
• Christian Bidard. 2012. The frail grounds of the Ricardian dynamics.
• Christian Bidard. 2013a. Intensive rent and value in Ricardo. Centro Sraffa.
• Christian Bidard. 2013b. Getting rid of rent.
• Christian Bidard. 2014. The Ricardian rent theory: An overview. Centro Sraffa working papers no. 8
• Christian Bidard. 2018. Ricardo and Ricardians on the order of cultivation. Journal of the History of Economic Thought 40 (3): 389-399.
• Guido Erreygers. 1995. On the uniqueness of cost-minimizing techniques. The Manchester School 63: 145-166.
• Heinz Kurz and Neri Salvadori. 1995. Theory of Production. Cambridge University Press.

Socialism Works As Participatory Budgeting

Socialists are not waiting for the revolution. They have started building socialism here and now.

Participatory budgeting is one process for providing most people in a city or municipality with more power to decide on city budgets. I do not fully understand it, but it includes more direct democracy. I suppose this process is close to how syndicates and soviets were supposed to work.

I think of a budget as an expression of values. Participatory budgeting is an attempt to allow all of a community to express their values, not just an elite few.

Participatory budgeting was first implemented in Porto Alegre, Brazil, in 1989. More about it is here and here. I see that they do it in Saratoga Springs, not too far away from where I am. As I understand it, New York City limits the part of their budget that follows the participatory budgeting process. But they do have some.

Anyways, participatory budgeting can be seen as a forerunner of what may come. Many know more than me on this topic.

A Switch Point Without Intersecting Wage Curves

Figure 1: Start of Wage Curves, with One Real and One Fake Switch Point

1.0 Introduction

This post presents another numeric example with pure fixed capital and extensive rent. Aside from these aspects of the model, no joint production exists.

Models of pure fixed capital or of extensive rent share certain properties with models of the production of commodities with labor and circulating capital alone. This article demonstrates that a model that combines pure fixed capital and extensive rent can exhibit issues raised by joint production. The cost-minimizing technique need not maximize the wage, and the choice of technique cannot be analyzed by the construction of the wage frontier. A switch point can exist without an intersection of wage curves, and intersections of wage curves can be fake switch points.

2.0 Technology

The example is specified by the technology, endowments of land, and requirements for use. An analysis of quantity flows identifies which techniques are feasible at a given level of requirements for use. The analysis of the choice of technique requires the examination of the solutions to the price systems for each technique.

I assume the existence of two types of land. More than one type is required for this model to exhibit extensive rent. With only two types of land, contrasting the orders of efficiency and of rentability is uninteresting. The order of efficiency is the order in which different types of land are introduced into cultivation as net output expands. The order of rentability sorts the lands by rent per acre. When both types of land are farmed, one type will be only partially farmed. It has a rent of zero; the other type of land obtains a positive rent. The orders of efficiency and rentability are necessarily identical, with two types of land and only one scarce. These orders can be completely reversed in models with more lands and both extensive and intensive rent.

Fixed capital is another aspect of joint production, in addition to land, in this model. A newly produced machine can be used for three years in production. Machines are assumed not to be consumption goods. New machines, but not old machines, can be consumer goods in models of pure fixed capital. This model seems to be close to the minimal complexity to investigate a combination of land-like natural resources and fixed capital in a model with the production of multiple commodities that is otherwise of single production alone. In a simpler model, the physical life of the machine would be only two years.

Table 1: Inputs for Processes Comprising the Technology

Input	Processes
	I	II	III	IV	V	VI	VII
Labor	a0,1 = 0.4	a0,2 = 0.2	a0,3 = 0.6	a0,4 = 0.4	a0,5 = 0.23	a0,6 = 0.59	a0,7 = 0.39
Type 1 Land	0	c1,2 = 1	c1,3 = 1	c1,4 = 1	1	1	1
Type 2 Land	0	0	0	0	c2,5 = 1	c2,6 = 1	c2,7 = 1
Corn	a1,1 = 0.1	a1,2 = 0.4	a1,3 = 0.578	a1,4 = 0.6	a1,5 = 0.39	a1,6 = 0.59	a1,7 = 0.61
New Machines	0	1	0	0	1	0	0
Type 1 1-Yr. Old Machines	0	0	1	0	0	0	0
Type 1 2-Yr. Old Machines	0	0	0	1	0	0	0
Type 2 1-Yr. Old Machines	0	0	0	0	0	1	0
Type 1 2-Yr. Old Machines	0	0	0	0	0	0	1

The technology is specified by the coefficients of production for seven processes. Each column in Table 1 shows the person-years of labor, acres of either type of land, bushels of corn, and numbers of new and old machines required as inputs to operate a process at unit level. The outputs of corn and machines, new and old, per unit level of each process are shown in Table 2. Machines are an industrial product which needs no land to produce. The laborers produce corn on land from inputs of corn and machines. Old machines one year older are produced jointly with corn from inputs of machines. Each old machine is of a type customized to the land on which it was produced. Old machines cannot be transferred from one type of land to another. They are assumed to be capable of free disposal. Formally, free disposal of an old machine of, say, type 1 is specified by assuming the existence of another process duplicating the second or third process, but without an output of an old machine. Each process is assumed to exhibit constant returns to scale (CRS) and to require a year to complete. The coefficients of production for the first four processes, other than those for land, are taken from a reswitching example by Baldone (1980).

Table 2: Outputs for Processes Comprising the Technology

Input	Processes
	I	II	III	IV	V	VI	VII
Corn	0	b1,2 = 1	b1,3 = 1	b1,4 = 1	b1,5 = 1	b1,6 = 1	b1,7 = 1
New Machines	1	0	0	0	0	0	0
Type 1 1-Yr. Old Machines	0	1	0	0	0	0	0
Type 1 2-Yr. Old Machines	0	0	1	0	0	0	0
Type 2 1-Yr. Old Machines	0	0	0	1	0	0	0
Type 1 2-Yr. Old Machines	0	0	0	0	0	1	0

The specification of model parameters is completed with endowments and requirements for use. Assume 100 acres of each type of land exist. The required net output is assumed to be 87 bushels corn. This required net output is such that all and only the techniques which require both types of land to be farmed are feasible.

3.0 Techniques and Feasibility

A technique is defined by which processes are operated, which type of lands are left unfarmed, which are partially farmed, and which are farmed to the full extent of their endowment. Rents can only be obtained on the last. Twenty-four techniques (Table 3) are defined for this technology. The capital goods that are used up in operating a technique can be reproduced. A net output remains, consisting, in the example, solely of corn.

Only scarce lands obtain a rent, and which are scarce varies with the technique. No land is scarce in the Alpha through Zeta techniques. One land is farmed and not to its full extent. Type 1 land is scarce in the Eta through Omicron techniques, while type 2 land is scarce in the remaining nine techniques. The techniques also vary in the economic life of the machine, one, two, or three years, on each type of land. Under the assumptions, the first six techniques are infeasible. Only Eta through Omega are feasible.

Table 3: Techniques of Production

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

4.0 The Price System

The modeled economy consists of three classes: workers, landlords, and capitalists. Capitalists buy inputs and hire workers who they direct to produce commodity outputs. In agriculture, capitalist farmers pay rent on scarce land to landlords. The capitalists choose the processes to operate based on cost. Accordingly, prices must be analyzed.

A system of equations is associated with each technique. An equation characterizes the prices for each process operated under a technique. These equations show the same rate of accounting profits is obtained on the value of the capital goods advanced at the start of the year. Rent and wages are paid out of the surplus product at the end of the year. A bushel corn is numeraire. The rent per acre appears in the equation for processes operating on the land that is fully farmed, if any. This land is scarce. Lands that are not fully farmed are free, and no rent appears in the equations for the processes operating on them.

5.0 On the Solutions of the Price Systems

Given the rate of profits, the price system for each technique can be solved. The solution for a technique has one degree of freedom. The solution can be presented with the wage, the price of new and old machines, and rents per acre as functions of the rate of profits. Figure 1 graphs the start of the wage curves for each technique in the example. Notice that the ordinate does not begin at zero in the graph. In this example, each wage curve is downward-sloping. Wage curves can be upward-sloping off the outer wage frontier in models of fixed capital. In this example with fixed capital and extensive rent, the wage frontier is neither the outer frontier of all wage curves nor the inner frontier.

In the illustrated range of the rate of profits, the wage frontier is the wage curve for the Zeta, Nu, Xi, and Omicron techniques. The wage curve for a technique is found from solving the price system formed from the machine-building process and the corn-producing processes operating on the non-scarce type of land. Quadrio Curzio & Pellizzari (2010) call this the ‘solving subsystem’. The Zeta, Nu, Xi, and Omicron techniques differ on which processes are operated on Type 1 land, but not on Type 2 land, which is free for all four techniques. Thus, they have the same solving system and the same wage curve.

Why are the wage curves for Nu and Omicron cost-minimizing in the illustrated range of the rate of profits? A technique is cost-minimizing at a given rate of profits if:

• The wage and the prices of all produced commodities (corn and machines of various types and vintages) are positive.
• The rent of the scarce type of land is positive.
• The prices of old machines not produced by the technique are negative for the price systems in which they are produced. Bidard (2016) defines ghost commodities as such non-produced commodities that affect the prices of produced commodities.

The price of a Type 1 old machine is negative under Omicron prices for rates of profits smaller than at the switch point between Nu and Omicron. A more general model would have processes that do not result from extending the economic life of a machine produced by the technique under consideration. For the technique to be cost-minimizing, no extra profits can be obtained by operating additional processes at the prices for the given rate of profits.

Two techniques are cost-minimizing at a switch point, except in fluke cases. The wage and the prices of all commodities produced with both techniques do not vary between the price systems for the two techniques. The rent per acre of land is also the same for the two techniques cost-minimizing at a switch point. Two types of switch points exist in the example, in addition to fake switch points.

In the first type, the techniques that are cost-minimizing for a switch point differ in the economic life of a machine. For example, the economic life of a machine used in farming Type 1 land is one year under Nu and three years under Omicron. Figure 2 illustrates the switch point between Nu and Omicron. Gamma, Sigma, Phi, Omega, Iota, Mu, and Omicron have positive prices for Type 1 one-year old machines in the graphed ranges of the rate of profits. Type 1 one-year old machines are also produced in the Beta, Rho, Upsilon, Psi, Theta, Lambda, and Xi techniques. Their prices are negative for these techniques in the indicated range. The price is zero, at the switch point, of the machine one year older than used in the technique with the shorter life in the price system for the other technique. A price of zero is a signal that the economic life of the machine can be truncated.

Figure 2: Price of Type 1 One-Year-Old Machines (Detail)

Rents per acre are zero at the other type of switch point. In the example, a switch point between Iota and Sigma exists at a rate of profits of approximately 45.04 percent. Their wage curves intersect at the switch point. The machine is run for its full physical life on Type 1 land under both techniques, and truncated after its first year of operation on Type 2 land. The techniques differ in which type of land is fully farmed and which is free. Figures 3 and 4 depict the rent curves for the example. The rent curve for Iota intersects the abscissa in Figure 3. Type 1 land is free under Sigma and has a rent per acre of zero under Iota at the switch point. Likewise, the rent curve for Sigma intersects the abscissa at the switch point in Figure 4. The rent per acre on Type 2 land is zero at the switch point.

Figure 3: Rent On Type 1 Land

Figure 4: Rent On Type 2 Land

A fluke switch point in which four techniques are cost-minimizing can combine these two types of switch points. Two techniques can differ in both the economic life of a machine and in which land is fully farmed. Two other techniques would then be cost-minimizing so that firms are indifferent between the economic life of the machine and which land is fully farmed. Two of these four techniques would differ in the economic life of a machine on scarce land; they would have the same wage curve. A switch point in which both the economic life of a machine and which type of land is scarce vary is the intersection of three wage curves.

Fake switch points arise when only two wage curves intersect for techniques which vary in both the economic life of a machine and the type of land that is fully farmed. Two fakes (Table 4) appear in the example. In both fakes, the prices of commodities produced under both techniques with intersecting wage curves do not vary between the techniques. For the first fake, the technique Omicron with the longer economic life of a machine is cost-minimizing. For the second fake, the technique Lambda with the longer economic life of a machine is not cost-minimizing. No price of these commodities not produced under both techniques are not zero under the technique in which they are produced. Their prices deviate from their behavior under the first type of switch point described above. On the other hand, the rent of one type of land, Type 2 for the first fake and Type 1 for the second, is zero for both techniques, as in the second type of switch point. The rent on the other type of land is positive for the technique for which it is scarce. The first switch point is a fake because the wage curve for Omega does not intersect with the other wage curves. Under Omicron and Omega, the economic lives of the machines are the same. The techniques differ in which land is scarce. By the same logic, the wage curve for Theta must intersect at the second switch point in Table 5 for it not to be a fake.

Table 4: Rent Per Acre Varies with the Technique at Fake Switch Points

Rate of Profits (Percent)	Technique	Commodities Produced Under Both	Ghost Commodities	Type 1 Land	Type 2 Land
15.9	Omicron*	Corn, New machines, Type 2 one and two-year old machines.	Type 1 one and two-year old machines. Prices of both are positive.	Scarce. Rent is positive.	Free
	Chi	Prices are positive and same as Omicron.		Free	Scarce. Rent is positive.
56.7	Lambda	Corn, new machines, Type 1 one-year old machines.	Type 1 one and two-year old machines. Prices of both are positive.	Scarce. Rent is positive.	Free
	Rho*	Prices are positive and same as Lambda.		Free	Scarce. Rent is positive.

6.0 The Cost-Minimizing Systems

A numeric example that combines the production and use of fixed capital with extensive rent is developed above. Table 5 summarizes the variation in the cost-minimizing technique through the full range of the rate of profits. The boundaries on the ranges at which techniques are cost-minimizing are approximate. The switch point between Pi and Rho exhibits capital-reversing. A higher wage or lower rate of profits is associated with the adoption of a technique that requires greater employment per unit of net output. This result is a challenge for what some obdurate economists still teach, that, under ideal assumptions, equilibria in the labor market must be the intersections of well-behaved, monotonic supply and demand curves. These results are also a challenge for claims by economists of the Austrian school. For the switch points between Iota and Omicron and between Rho and Sigma, a longer economic life of a machine is associated with greater capital-intensity, as they would expect. But for the switch points between Nu and Omicron and between Pi and Rho, a shorter economic life of a machine is associated with greater capital-intensity

Table 5: Cost-Minimizing Techniques

Range (Percent)	Technique	Economic Life of Machine (Years)		Land
		Type 1	Type 2	Type 1	Type 2
0 ≤ r ≤ 5.12	Nu	1	3	Scarce	Free
5.12 ≤ r ≤ 36.3	Omicron	3	3	Scarce	Free
36.3 ≤ r ≤ 45.0	Iota	3	1	Scarce	Free
45.0 ≤ r ≤ 55.7	Sigma	3	1	Free	Scarce
55.7 ≤ r ≤ 62.7	Rho	2	1	Free	Scarce
62.7 ≤ r ≤ 74.2	Pi	1	1	Free	Scarce

7.0 Conclusions

Joint production presents the possibilities of many phenomena inconsistent with clear properties of models of the production of commodities with circulating capital alone. This article demonstrates that at least some of these phenomena can occur with the combination of fixed capital and extensive rent, even though they do not occur in models of pure fixed capital and extensive rent considered separately. The choice of technique cannot be analyzed solely by the construction of the wage frontier. A switch point exists at which two wage curves do not intersect. Two fake switch points exist in the example, where rents per acre are not equal on one type of land at the switch point for the techniques with intersecting wage curves. The feasible technique with the largest wage is not necessarily cost-minimizing

No claim is made that other issues of joint production might not arise in models combining fixed capital and extensive rent. D’Agata (1983) provides an example in a model of intensive rent with a non-unique and sometimes upward-sloping wage frontier. The model in this article is similar to a model of intensive rent in some ways. Can an example be given with these properties?

A model with more types of land provides a setting for comparing and contrasting the orders of efficiency and rentability. The analysis in this article demonstrates that the wage frontier for cost-minimizing techniques is disconnected from the ordering of wage curves. How does the order in which lands are introduced into cultivation, at a given rate of profits, relate to the ordering of wage curves in models with fixed capital? Presumably, the introduction of fixed capital still allows for the order of rentability to differ from the order of efficiency. More efficient lands are not necessarily paid a higher rent per acre.

Models of rent emphasize the need to consider technical change. Net output can be increased only up to a hard limit. The introduction of new processes and techniques, a capability to extend the physical life of machines, the discovery of new natural resources, or decreases in some coefficients of production for existing processes are required to increase net output beyond that limit. Introduction of such possibilities into the model will result in structural economic dynamics.

Neither Socialist Nor Pro-Capitalist In The Underworld

This post briefly describes some advocates of some strange ideas near the start of the twentieth century. I am avoiding socialists and Marxists in what Keynes called an "underworld" and an "army of heretics".

• Major C. H. Douglas: British engineer who inspired the Canadian social credit movement. I have not read enough to understand his A + B theorem, but I gather he explained depressions and recessions by an underconsumptionist theory.
• William Trufant Foster and Waddill Catchings: American writers and underconsumptionists. Hayek wrote some criticisms of them.
• Silvio Gesell: A german living in Argentina and later in the Soviet cabinet in Bavaria after World War I. Had a worldwide following. Advocated stamped money, in which money needed to be stamped each month to remain capable purchasing commodities. Money would no longer be as liquid. Savings would be more likely to be channeled into physical investments.
• J. A. Hodgson: Birtish journalist and prolific popular writer. Developed a theory of underconsumption, rejecting Say's law. His theory of imperialism influenced later Marxists.
• Henry George: American journalist and writer. His book Progress and Poverty started a worldwide movement, with followers today. Most well-known for the idea of a land value tax (LVT).
• Frederick Soddy: English chemist, 1921 Nobel laureate. Had ideas about how thermodynamics applied to economics. Advocated the abolishment of the gold standard. Claimed that debts must grow exponentially with compound interest, which cannot be supported by the real economy.

I am sure that some of these were associated with some political ideas I would reject. Wikipedia has Major Douglas and Frederick Soddy as anti-semites.