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.