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.

Helen McCabe On 'Capitalism'

If you want to change a system, it is helpful to have a name for it. Apparently, Louis Blanc created the word 'capitalism':

"It is worth noting here that Mill never uses the term 'capitalism,' though later commentators have been happy to use it on his behalf in describing his views. Blanc is generally credited with the first use of the term in our modern meaning (in his Pages de l’histoire de la révolution de 1848, 1850), so Mill could have come across the term via his friendship with Blanc (and his reading of Blanc's work, though we have no evidence he read this particular one). Similarly, Ricardo had used the term 'capitalist,' and Mill was very familiar with his work. Where Mill uses 'capitalist,' it is in a similar sense - referring to a person who owns capital. His examples in Principles are all of individual manufacturers who own factories, employ labourers, and are involved in the day-to-day running of their business, despite spending some of their income 'in hiring grooms and valets, or maintaining hunters and hounds,' which implies they also have a good deal of leisure. This is also similar to the way in which Coleridge (in 1823), Proudhon (in 1840), and Disraeli (in 1845) use the word. Even Marx only began using 'capitalism' (rather than 'the capitalistic system') in Capital (in 1867), and generally preferred 'the capitalist mode of production.'" -- Helen McCabe, John Stuart Mill, Socialist (2021)

As can be seen above, the word 'capitalist' predates 'capitalism'. The word 'capital' is much older. Adam Smith talked about 'commercial society'. Friedrich Hayek talked about the 'Great Society'. Hegel, in the Philosophy of Right, wrote about 'civil society', a term later taken up by, among others, Gramsci.

Persistent Unemployment: A Flaw In Capitalism

Defenders of capitalism, whether of a liberal reformist or of a more conservative variety, portray it as harmonious, in some sense. One aspect of this portrayal is that under ideal conditions, at least, markets tend to clear. This supposed tendency must apply especially to the labor market. Another aspect, put aside in this post, is the obsolete and explopoded marginal productivity theory of distribution.

Nicholas Wapshott was correct in putting this supposed tendency as a central point of contention between John Maynard Keynes and Friedrich Hayek:

"... the objections of Circus members ... suggested to Keynes what was to become a pivotal element of The General Theory, that overall output was not fixed and could be raised through increased investment to a point where everyone in an economy was employed. It was this first slender thread of thought that led to Keynes's wholesale contradiction of the claim of classical economists like Hayek that an economy, left to its own devices, in the long run inevitably came to rest at a state of equilibrium where there was full employment. Keynes was to argue in The General Theory that in the short and medium terms an economy could reach equilibrium with considerable unemployment and that the full employment equilibrium predicted by classical economists too often proved to be elusive. Keynes believed that the chronic unemployment endured in Britain and America in the 1920s and 1930s was evidence that the full employment equilibrium was a fallacy." -- Nicholas Wapshott. 2011. Keynes Hayek: The Clash That Defined Modern Economics. New York: W. W. Norton and Co.: p. 128.

I can think of three positions to take here:

• No such tendency for markets, including labor markets, to clear in any run.
• Such tendency would exist, but imperfections and rigidities prevent this tendency from manifesting itself.
• Although imperfects and such like exist, they are not empirically important enough to prevent this tendency for markets to clear, if we are patient enough.

Keynes and Piero Sraffa have convincingly argued for the first position, that no tendency exists for markets, including the labor market to clear. Keynes had many arguments. For example, workers and capitalists may have no way of negotiating about the real wage. Suppose unemployment is high. A system-wide drop in the money wage occurs. Prices might drop as a result, leaving the real wage unchanged. If the money supply is exogenous, the real money supply is increased in some sense. You might get money policy implemented by trade unions. But this effect is outside the usual partial equilibrium story about the labor 'market'. Furthermore, capitalists would be inclined not to open new factories in a period of declining wages and prices. They can anticipate that new factories opened tomorrow, instead of today, will have decreased costs and a decreased burden of debt. With declining prices of output, too, capitalists will put off investment.

Sraffa showed that prices, in a long run theory, do not have the properties needed to support just-so stories about supply and demand. Prices of production can support a long run theory in which labor markets do not necessarily clear. Say's law is false.

For the second position, I think of failures of competition, information asymmetries, search costs, incomplete contracts, principal agent problems, and so on. Many misleadingly characterize Keynes as holding this position. I guess those investigating these imperfections often advocate government interventions to remove or mitigate their effects. A radical position is that at least some of these imperfections cannot be removed. Government might then need to simulate how markets would work without these limitations. Is this the position of ordoliberalism?

Those with the third position would be for a more laissez faire approach.

But to return to the first position. Suppose no tendency exists for markets, including the labor market, to clear. Keynes recommended a "somewhat comprehensive socialisation of investment". But he was not clear on what that means.

Fake Switch Points With Fixed Capital And Extensive Rent

Figure 1: An Enlargement Of Wage Curves

1.0 Introduction

This long post is a start at addressing this problem statement. With a bit of improvement on the scholarship in the introduction and an appendix presenting the solutions of the price systems, it would be an article to be submitted to a journal. But I will not do this before exploring other examples. I would like to have an upward-sloping wage frontier, in particular.

Models of the production of commodities with circulating capital have certain nice properties. Models of pure fixed capital and of extensive rent have the same nice properties, for the most part. This article contends that a model that combines pure fixed capital and extensive rent need not have these properties. Many of the issues that arise in models of general joint production also arise in a model combining fixed capital and extensive rent.

The choice of technique can be analyzed in models with circulating capital alone by constructing the outer envelope of the wage curves for each technique. Each wage curve slopes down. The wage, for a technique, is lower the higher the rate of profits. The cost-minimizing technique at a given rate of profits is unique, except at switch points. The "determination of the cost-minimising technique is independent of the structure of requirements for use" (Huang 2019). The wage and prices of production are unique functions of the rate of profits. If a feasible technique exists with a defined wage and prices of production at a given rate of profits, then a cost-minimizing technique exists. A market algorithm (Bidard 1990 and Vienneau 2017) converges, without going into a cycle.

None of these properties are necessarily true in a general system of joint production.

Models with fixed capital are special cases of models of joint production. Bidard (2004), Kurz & Salvadori (1995), Pasinetti (1980), Schefold (1989) and Woods (1990) develop the theory of pure fixed capital. Huang (2019) is a survey. In models of pure fixed capital, the choice of technique can be analyzed by constructing the outer frontier of the wage curves for the various techniques.

In the simplest model of extensive rent, a single agricultural commodity is produced, on each type of land, with a single production process. All types of land, except for one, are farmed to the full extent of their endowment. No alternate processes are available on any type of land for producing the agricultural commodity. Prices of production are such that a single rate of profits is made in all operated processes. Rent is obtained by landlords who own the scarce lands. The type of land that is only partially farmed is not scarce and does not pay a rent.

At a given rate of profits, the techniques in a model of extensive rent can be ordered by the wage. This is the order of efficiency, also known as the order of fertility. The cost-minimizing technique is the first technique, in decreasing order of wages, that is feasible and, away from switch points, has positive rents on all lands that are fully farmed. In this sense, the choice of technique can be analyzed by the construction of the wage frontier in models of extensive rent.

The choice of technique cannot generally be analyzed by the construction of the wage frontier in models that combine fixed capital and extensive rent. In particular, an example is given with a fake switch point. The wage curve on the frontier intersects another wage curve. Yet that point of intersection is not a switch point.

2.0 Numeric Example of the Model

This article presents a model combining fixed capital and extensive rent, by a numerical example. The numerical example is specified by the definition of 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 system for the techniques.

2.1 Technology, Endowments, and Requirements for Use

The parameters for the model specify the technology, endowments of land, and requirements for use. 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 kinds 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. These orders can be in completely reversed order 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 two years in production. Machines are assumed not to be consumption goods. In models of pure fixed capital, new machines but not old machines can be consumer goods. This model seems to be of 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.

The technology is specified by the coefficients of production for five 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 are produced jointly with corn from inputs of new 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 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 three processes, other than those for land, are taken from a reswitching example (Schefold 1980).

Table 1: Inputs for Five Processes Comprise the Technology

Input	Industry
	Machine	Corn
	Process I	Process II	Process III	Process IV	Process V
Labor	a0,1 = 1/10	a0,2 = 43/40	a0,3 = 1	a0,4 = 1	a0,5 = 43/40
Type 1 Land	c1,1 = 0	c1,2 = 1	c1,3 = 1	c1,4 = 0	c1,5 = 0
Type 2 Land	c2,1 = 0	c2,2 = 0	c2,3 = 0	c2,4 = 1	c2,5 = 1
Corn	a1,1 = 1/16	a1,2 = 1/16	a1,3 = 1/4	a1,4 = 1/16	a1,5 = 3/10
New Machines	a2,1 = 0	a2,2 = 1	a2,3 = 0	a2,4 = 1	a2,5 = 0
Type 1 Old Machines	a3,1 = 0	a3,2 = 0	a3,3 = 1	a3,4 = 0	a3,5 = 0
Type 2 Old Machines	a4,1 = 0	a4,2 = 0	a4,3 = 0	a4,4 = 0	a4,5 = 1

Table 2: Outputs for Five Processes Comprise the Technology

Input	Industry
	Machine	Corn
	Process I	Process II	Process III	Process IV	Process V
Corn	b1,1 = 0	b1,2 = 1	b1,3 = 1	b1,4 = 6/5	b1,5 = 4/5
New Machines	b2,1 = 1	b2,2 = 0	b2,3 = 0	b2,4 = 0	b2,5 = 0
Type 1 Old Machines	b3,1 = 0	b3,2 = 1	b3,3 = 0	b3,4 = 0	b3,5 = 0
Type 2 Old Machines	b4,1 = 0	b4,2 = 0	b4,3 = 0	b4,4 = 1	b4,5 = 1

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 anywhere from 107.5 bushels of corn to 160 bushels. A required net output in this range, but not at the endpoints, is such that all and only the techniques which require both types of land to be farmed are feasible.

2.2 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. Twelve 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.

Table 3: Techniques of Production

Technique	Processes	Land
		Type 1	Type 2
Alpha	I, II	Partially farmed	Fallow
Beta	I, II, III	Partially farmed	Fallow
Gamma	I, IV	Fallow	Partially farmed
Delta	I, IV, V	Fallow	Partially farmed
Epsilon	I, II, IV	Fully farmed	Partially farmed
Zeta	I, II, III, IV	Fully farmed	Partially farmed
Eta	I, II, IV, V	Fully farmed	Partially farmed
Theta	I, II, III, IV, V	Fully farmed	Partially farmed
Iota	I, II, IV	Parially farmed	Fully farmed
Kappa	I, II, III, IV	Parially farmed	Fully farmed
Lambda	I,II, IV, V	Parially farmed	Fully farmed
Mu	I, II, III, IV, V	Parially farmed	Fully farmed

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

2.3 The Price Systems

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, farmers pay rent on scarce land. The capitalists choose the processes to operate based on cost. Accordingly, prices must be analyzed.

A system of equations is associated with each technique. As an example of a technique, consider Kappa. The following four equations present its price system:

a_1,1(1 + r) + w_κ(r) a_0,1 = p_1,κ(r)

[a_1,2 + p_1,κ(r)](1 + r) + w_κ(r) a_0,2 = b_1,2 + p_2,κ(r)

[a_1,3 + p_2,κ(r)](1 + r) + w_κ(r) a_0,3 = b_1,3

[a_1,4 + p_1,κ(r)](1 + r) + rho_2,κ(r) c_2,4 + w_κ(r) a_0,4 = b_1,4

Table 4 defines the price variables. 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.

Table 4: Functions for Solutions for Kappa Technique

Function	Defintion
p1,κ(r)	The price of a new machine, in bushels per machine.
p2,κ(r)	The price of an old type 1 machine, in bushels per machine.
p3,κ(r)	The price of an old type 2 machine, in bushels per machine.
rho1,κ(r)	The rent of type 1 land, in bushels per acre.
rho2,κ(r)	The rent of type 2 land, in bushels per acre.
wκ(r)	The wage, in bushels per person-year.

Under Kappa, machines are operated for their full physical life in farming type 1 land. Accordingly, the price of old type 1 machines appears in the price equations, with process II producing old machines jointly with corn. Machines are discarded after one year in farming type 2 land. Consequently, the price of type 2 old machines is zero and their price does not appear in the price equations in Display 1.

Type 2 land is fully farmed under Kappa. It is scarce, and its rent appears in the above equations. Type 1 land, on the other hand, is not scarce. Its rent is zero. The following display expresses that type 2 machines are not used and that that type 1 land is free:

p_3,κ(r) = 0, rho_1,κ(r) = 0

2.4 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 consists of the functions listed in Table 4. Figure 2 graphs the wage curves for each technique in the example; Figure 1 is an enlargement. The first two intersections of wage curves in Figure 1 cannot be distinguished in Figure 2. The wage frontier is composed of the wage curves for the cost-minimizing techniques. Section 3 demonstrates that the Iota and Kappa techniques are cost-minimizing in the indicated ranges of the rate of profits. 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 the example with fixed capital and extensive rent, the wage frontier is neither the outer frontier of all wage curves nor the inner frontier. The wage frontier need not be always downward-sloping in other examples.

Figure 2: Wage Curves

The solutions of the price systems also provide rent curves, rent per acre, as a function of the rate of profits. Figure 3 plots the rent curves for the four techniques in which rent is obtained on type 1 land. Only the first quadrant is shown. The rent curves for Epsilon and Zeta lie entirely below the abscissa. Rent is zero for Eta at the intersection of the rent curves for Alpha and Delta. That is, the cost of producing corn is the same with process II or a combination of processes IV and V at this rate of profits. By the same logic, rent is zero for Theta at the intersection of the wage curves for Beta and Delta.

Figure 3: Rent on Type 1 Land

Figure 4 shows the rent curves for the four techniques in which rent is obtained on type 2 land. The zero for the rent curve for Lambda is at the rate of profits at which the wage curves for Alpha and Delta intersect. The zero for the rent curve for Mu is at the fake switch point. These zeros cannot be distinguished by eye in Figure 4. All prices are the same for both techniques at switch points between Iota and Kappa. Thus, the rent curves for type 2 land intersect for these techniques at switch points.

Figure 4: Rent on Type 2 Land

Table 5 summarizes the claims about the variation in the cost-minimizing technique with the rate of profits. Under Kappa, the machine is operated for its full physical life on type 1 land, but only for one year on type 2 land. Type 2 land is scarce. Under Iota, the machine is discarded after one year, no matter which type of land it is operated on. Type 2 land remains scarce. This is a reswitching example. Kappa is adopted at low and high rates of profits. The switch point at a rate of profits of 50 percent is an example of capital-reversing.

Table 5: Cost-Minimizing Techniques

Region	Range	Technique	Old Machines	Rents
1	0 < r < 1/3	Kappa	p2,κ > 0, p3,κ = 0	rho1 = 0, rho2 > 0
2	1/3 < r < 1/2	Iota	p2,ι = 0, p3,ι = 0
3	1/2 < r < 258.8%	Kappa	p2,κ > 0, p3,κ = 0

3.0 Results

The solutions of the price equations can be synthesized to justify claims about the cost-minimizing techniques. In models of extensive rent, the cost-minimizing technique is found, at a given rate of profits, by working downwards through the wage curves until one is found, with non-negative rents, for a feasible technique. That method does not work in this example with fixed capital.

Table 6: Some Properties of Feasible Techniques

Technique	Old Machines	Rent
Epsilon	Not produced	rho1,ε < 0, for all r
Zeta	p2,ζ > 0, for all r	rho1,ζ < 0, for all r
Eta	p3,η < 0, for all r	rho1,η > 0, for r < 24.4%
Theta	p2,θ > 0, p3,θ < 0, for all r	rho1,θ > 0, for r < 24.8%
Iota	Not produced	rho2,ι > 0, for all r
Kappa	p2,κ > 0, for r < 1/3 or r > 1/2	rho2,κ > 0, for all r
Lambda	p3,λ < 0, for all r	rho2,λ > 0, for r > 24.4%
Mu	p2,μ > 0, for r < 1/3 or r > 1/2; p3,μ < 0, for all r	rho2,μ > 0, for r > 24.8%

The wage frontier is composed of the wage curves for the cost-minimizing techniques. The wage frontier for this example of extensive rent is neither the outer frontier nor the inner frontier of the wage curves. A wage curve for a technique contributes to the frontier, at a given rate of profits, only if all old machines produced by the technique have non-negative prices and the rent on land scarce under the technique is non-negative. Only the wage curves for Iota and Kappa satisfy these criteria (Table 6)

Figure 5: Iota Cost-Minimizing at Middling Rates of Profits

Under Iota, process II is operated on type 1 land, and process IV is operated on type 2 land. Type 2 land is fully farmed and obtains a rent. Iota is cost-minimizing if processes III and V do not pay extra profits. Extra profits are the difference between revenues and costs, where advances of capital goods are costed up at the going rate of profits. The following equations define extra profits per unit level of operation for these processes under Iota prices:

Π_{III, ι} = b_1,3 - [a_1,3(1 + r) + w_ι a_0,3]

Π_{V, ι} = b_1,5 - [a_1,5(1 + r) + rho_2,ι c_2,5 + w_ι a_0,5]

As shown in Figure 5, capitalists obtain extra profits by operating process III when Iota prices rule at low and high rates of profits. In other words, capitalists will extend the economic life of the machine when farming type 1 lands.

Figure 6: Extra Profits Not Available at Kappa Prices by Extending Life of Machine

Process V is the only process that is not operated in the Kappa technique. Extra profits in process V, at unit level under Kappa, are defined by the following equation:

Π_{V, κ} = b_1,5 - [a_1,5(1 + r) + rho_2,κ c_2,5 + w_κ a_0,5]

Figure 6 plots these extra profits, as a function of the rate of profits. They are always negative. An analysis of extra profits under Kappa cannot find switch points with Iota, in which the economic life of the machine is truncated on type 1 land. Figure 7 plots the price of type 1 old machines. This price turns negative under Kappa at the switch points with Iota. In models of fixed capital, a negative price of an old machine signals to managers of firms that cost can only be minimized if the economic life of a machine is truncated.

Figure 7: The Price of Type 1 Old Machines

At a switch point, the prices for the two techniques for which the wage curves intersect are the same. At the fake switch point between Kappa and Eta or Theta, the wage and the price of a new machine are invariant among the techniques. The price of a type 1 old machine is equal at a positive price between Kappa and Theta. An old machine is not produced under Eta. Thus, the fake switch point cannot be a switch point between Kappa and Eta. But the price of a type 2 old machine is negative under Theta, not zero. No extra profits are available under Kappa by extending the economic life of the machine in farming type 2 land. So the switch point is a fake. The rents are also inconsistent with the switch point being genuine. Rent on type 1 land is negative under Eta and zero under both Kappa and Theta. But rent on type 2 land is positive under Kappa, not zero, as under Eta and Theta.

In a model combining fixed capital and extensive rent the intersection of a wage curve with the wage frontier for the cost-minimizing technique can be a fake.

Conclusion

A model combining fixed capital and extensive rent can exhibit at least some of the difficulties in models of general joint production that do not arise with fixed capital and extensive rent, considered separately. A fake switch point exists in the reswitching example. A simple examination of wage curves does not reveal which technique is cost-minimizing. The feasible technique with the largest wage is not necessarily cost-minimizing.

References

• D'Agata, A. 1983. The existence and unicity of cost-minimizing systems in intensive rent theory, Metroeconomica, 35: 147-158.
• Baldone, Salvatore. 1974. Il capitale fisso nella schema teorico di Piero Sraffa. Studi Economici XXIX(1): 45-106 (Tr. In Pasinetti 1980).
• Bidard, Christian. 1990. An algorithmic theory of the choice of techniques. Econometrica 58(4): 839-859.
• Bidard, Christian. 1997. Pure joint production. Cambridge Journal of Economics 21(6): 685-701.
• Bidard, Christian. 2004. Prices, Reproduction, Scarcity. Cambridge: Cambridge University Press.
• Bidard, Christian and Edith Klimovsky. 2004. Switches and fake switches in methods of production. Cambridge Journal of Economics 28 (1): 88-97.
• Huang, B. 2019. Revisiting fixed capital models in the Sraffa framework. Economia Politica 36: 351-371.
• Kurz, Heinz and Neri Salvadori. 1995. Theory of Production: A Long-Period Analysis. Cambridge: Cambridge University Press.
• Pasinetti, Luigi L. (ed.). 1980. Essays on the Theory of Joint Production. New York: Columbia University Press.
• Quadrio Curzio, Alberto. 1980. Rent, income distribution, and orders of efficiency and rentability, (In Pasinetti 1980).
• Quadrio Curzio, Alberto and Fausta Pellizzari. 2010. Rent, Resources, Technologies. Berlin: Springer.
• Salvadori, Neri. 1999. Transferable machines with uniform efficiency paths. Value, Distribution and Capital: Essays in honour of Pierangelo Garegnani (ed. by G. Mongiovi and F. Petri). New York: Routledge.
• Schefold, Bertram. 1980. Fixed capital as a joint product and the analysis of accumulation with different forms of technical progress. (In Pasinetti 1980).
• Schefold, Bertram. 1989. Mr. Sraffa on Joint Production and other Essays. London: Unwin-Hyman.
• Sraffa, Piero. 1960. The Production of Commodities by Means of Commodities: A Prelude to a Critique of Economic Theory. Cambridge: Cambridge University Press.
• Varri, Paolo. 1974. Prezzo, saggio del profitto e durata del capitale fisso nello schema teorica di Piero Sraffa. Studi Economici XXIX(1): 5-44 (Tr. In Pasinetti 1980).
• Vienneau, Robert L. 2017. The choice of technique with multiple and complex interest rates. Review of Political Economy 29(3): 450-453.
• Woods, J. E. 1990. The Production of Commodities: An Introduction to Sraffa. Atlantic Highlands, NJ: Humanities Press International.

Catalog of Capital-Theoretic "Paradoxes"

This post lists capital-theory 'paradoxes' or 'perversities'. The label of 'paradox' or 'perverse' merely indicates that they are inconsistent with traditional marginalist theory. I think of the following:

• Reswitching of techniques: Out of a given book of recipes, a technique is cost-minimizing at two ranges of the rate of profits, with some other technique cost-minimizing in-between. A technique is a combination of processes, one for each industry, that characterizes production in the economy as a whole.
• Capital-reversing: A switch point is a rate of profits or wage at which two techniques are cost-minimizing. Around a switch point with capital-reversing, a lower rate of profits is associated with a technique being replaced with one with a lower capital-intensity. Or around such a switch point, a higher wage is associated with a higher labor intensity. You can say that a labor demand curve, for the economy as a whole, slope up.
• Reverse labor substitution: Around a switch point with reverse labor substitution, a higher wage is associated with the adoption of a process in a specified industry with more labor hired per unit gross output in that industry. You can say that a sectorial labor demand function slopes up.
• Process recurrence: In a specified industry, the same process can be in the cost-minimizing techniques at two different rates of profits, with a different process being in the cost-minimizing technique in-between.
• Recurrence of truncation: The cost-minimizing technique, in models of fixed capital, can require that an old machine be discarded before its physical life ends. The recurrence of truncation occurs when the same economic life of a machine is adopted in an industry at two different ranges of the rate of profits, with a different economic life in-between.
• Reswitching of the order of fertility: In models of rent, the order of fertility is the order in which different (types of) land are cultivated, at a given rate of pofits, as output expands. This order can be the same at two ranges of the rate of profits, with a different order in-between.
• Reswitching of the order of rentability: In models of rent, the order of rentability is the order, at a given rate of profits of lands when sorted by rent per acre. This order can be the same at two ranges of the rate of profits, with a different order in-between.
• Association of the lengthing of the economic life of a machine with smaller capital-intensity: The adoption of a longer economic life of a machine can be consistent with a smaller capital-intensity.
• Association of the adoption of a machine requiring a more roundabout process with smaller capital-intensity: The cookbook can have a choice of the use of different types of machines in an industry. The technique that is more roundabout, that is, with a machine that lasts longer, can have a smaller capital-intensity.

The relationships among these different phenomena are tedious to state.. The reswitching of techniques implies that capital-reversing occurs. But capital-reversing can occur without the reswitching of techniques. Reverse labor substitution can occur without either reswitching or capital-reversing. I guess process recurrence cannot occur without reverse labor substitution, but reverse labor substitution can occur without process recurrence. The recurrence of truncation is analogous to process recurrence, but somewhat different. The recurrence of trunctation implies that reverse labor substitution occurs. This recurrence is consistent with an absence of the reswitching of techniques. The reswitching of the order of fertility is consistent with no variations in quantity flows. Which lands are fully farmed and which are only partially farmed need not vary. Thus, the reswitching of the order of fertility does not imply the existence of capital-reversing. I hasten to add that the reswitching of techniques and capital-reversing can arise in models with rent.

The above list is not complete. For example, in models of international trade, a country can specialize as in the theory of comparative advantage, with less output than in autarky. Something like the reswitching of techniques can arise in models of spatial economics. Around a city, two areas at different distances from downtown might be specialized for agriculture, with an area specialized for industry in-between. (I haven't looked into this last case in any detail.)

Many economists may have never seen the economic theory that I draw on. But the above phenomena, which might be called Sraffa effects, follow from the assumptions of mainstream economics. In other words, this post describes some elements of modern economics.

Is The Order Of Efficiency Unique At A Given Rate Of Profits In Models With Multiple Agricultural Commodities?

1.0 Introduction

This post puts forth two research questions that I think I will never get to. And it puts forth one I have started on.

2.0 Is the Order of Efficiency Well-Defined with Mutliple Agricultural Commodities?

I have worked through an example with absolute, extensive, and intensive rent. The orders of efficiency and rentability are uniquely defined in this example. The example has an industrially produced commodity and one commodity, 'corn' produced on land.

Suppose more than one agricultural commodity exists. For example corn and barley can both be produced on land. But assume that no joint production, other than the use of land, exists. Would the order of efficiency still be uniquely defined at a given rate of profits? Would it depend on the composition of corn and barley in net output, as well as the level of net output?

I see even in my example, which techniques are feasible depend on the amount of iron and corn in net output. But would multiple agricultural commodities create more issues for defining the order of efficiency?

3.0 Can Competive Marginal Products Rank Inputs in the Opposite Order of their Productivity?

Frank Hahn argues that marginal productivity, rightly understood, is entirely consistent with Sraffa's model. He concentrated on the case of single production. with a discrete technology, the value of the marginal product of labor, for example, is an interval formed out of the right-hand and left-hand derivatives of production functions.

Hahn considers intertemporal equilibrium paths in which endowments are not given. Rather, they are found as part of the solution for steady state paths. I would rather take them as limit points of non-steady state paths, perhaps with saddle-point stability. But setting these points aside, I cannot see that Hahn is otherwise wrong. Marginal productivity by itself is not a theory of income distribution.

But how does marginal productivity apply to a combination of intensive and extensive rent? In my example, I find a range of the rate of profits in which three types of land are ordered by efficiency in exactly the opposite order as rent per acre. This does not seem consistent with J. B. Clark's idea that, with marginal products, you get rewarded for your contributions (or rather the contributions of the factors you own) to production.

4.0 Which Isues Arising From General Joint Production Arise In Models Combining Extensive Rent and Fixed Capital?

I give an example. In such models, the wage frontier for the cost-minimizing technique is not the outer fronter for wages curves. So I guess, it can be upward-sloping. Can the cost-minimizing technique be non-unique away from switch points? Can I find Bidard's market economy converging to a cycle, with no cost-minimizing technique existing? D'Agata has examples in a model of inetensive rent.