Two Ways To Think About Fixed Capital

1.0 Introduction

This post presents two approaches to fixed capital. The first emphasizes power and convention. It emphasizes that the 'money world', as Mason and Jayadev (2026) put it, is not a direct reflection of physical properties of production. The second is much closer to a determined model, with one degree of freedom.

I have at least one previous post, more abstractly on these themes

2.0 Depreciation as Accounting Conventions and Mediated by the Exercise of Power

Robert Paul Wolff points out that how overhead costs are allocated is a political exercise in many firms. Two managers on the same level might each be in charge of the production of a specific product. Suppose these products are produced in the facility. Each manager wants upper management to see that they are making the most profits for the company. In is in the interest of each to see that as much of possible of the overhead for running the facility is applied to their rival's department.

A given machine might be used in multiple production processes, resulting in the production of different products. The allocation of overhead costs here follows the above logic.

Sometimes (many times?), no fact of the matter exists for how to allocate costs. As I understand it, the Generally Accepted Accounting Principles (GAAP) includes different conventions for depreciation. Some models I have seen assume radioactive decay so to speak. Tax law specifies what can be allowed.

3.0 Depreciation and the Economic Life of Machines as an Aspect of Prices of Production

Another way of thinking about fixed capital is as in Sraffa (1960). Enough equations are given for prices to be determined, corresponding to a given wage or rate of profits. The data include inputs and outputs, in disaggregated physical terms, for all combinations of ages of machines. These detailed specifications are not based solely on the age, but the entire history of each machine.

Alessandro Roncaglia showed, back in the 1970s, that a countable infinity of equations can arise when production processes can require inputs of combinations of machines. The theory of pure fixed capital includes assumptions that rule out this possibility.

Sraffa derives the formula for an annuity from his treatment of a machine of constant efficiency. But the analysis generalizes. This approach is in tension with the first approach.

4.0 Conclusion

I will continue exploring mathematical puzzles. I like to think that Sraffa presents the elements of an alternative theory, not just an immanent critique of 'neoclassical' theory. The analysis points outside itself.

Reference

• Mason, J. W. and Arjun Jayadev. 2026. Against Money. University of Chicago.

On The Incoherence Of Austrian Business Cycle Theory

I thought I would try to summarize again some objections to the Austrian school.

Austrian Business Cycle Theory (ABCT) focuses on the consequences of the monetary authority setting the monetary interest rate below the natural rate of interest. Following Knut Wicksell somewhat, the theory argues that capitalist entrepreneurs will lengthen production processes. Since these decisions do not synchronize with household consumption and savings decisions, the artificial boom is unsustainable. A bust is the result.

This theory is built on mistaken capital theory. When Sraffa spanked Hayek, he deliberately put capital theory aside.

What would it mean for a production technique to be more capital-intensive? To examine the (il)logic of this approach, I make various simplifying assumptions.

Accordingly, consider vertically-integrated firms. They produce a given commodity or, rather, a basket of commodities, in fixed proportions. The only input is homogenous labor. All means of production, tools, intermediate goods are produced and used internally.

Under these assumptions, one can talk about (net) output per person-year. Productivity is well-defined. I start by postulating that with a more capital-intensive technique, workers are more productive.

It turns out that a lower interest rate does not induce managers of firms to adopt more capital-intensive techniques. Numerical examples illustrating this point have been available in the literature since the 1960s. And they are accepted by all sides. "The interesting point, however, is the perversity, not the duplicity." -- Robinson and Naqvi (1967).

I have now demonstrated that marginalist capital theory, including the Austrian variant, is invalid. Given typical assumptions, the traditional stories about 'capital' markets do not follow. But what about all that stuff Austrian school economists say about the structure of production?

They are wrong there, too. First, I consider aggregate measures of the period of production. Bohm-Bawerk's measure assumes simple interest, not compound interest. The counterexamples mentioned above demonstrate it is invalid to conclude cost-minimizing entrepreneurs will lengthen the period of production when they anticipate lower interest rates.

Nicolas Cachanosky and Peter Lewin have recently proposed a financial measure of Duration. By this measure, the technique chosen at a lower interest rate, around a switch point, has a larger Duration. But a larger Duration is associated with lower productivity in the counter-examples.

You could also consider how long capitalists will choose to run given machinery. Machines last for more than one production cycle, and capitalists must choose to set their economic life. Surely, a lower interest rate will provide incentives to capitalists to increase their economic life. Well, no. A longer economic life of a machine can be associated with a less capital-intensive technique in the sense that the productivity of labor is decreased. I happen to know that the recurrence of the period of truncation is possible without the reswitching of techniques.

And then there are Hayekian triangles. They do not work either. Hayekian triangles, as Roger Garrison notes, are heuristic pictures, useful for pedagogic purposes. Hayek unsuccessfully tried to put them on rigorous foundations in Prices and Production. But it all fell apart. He even discovered capital-reversing, in some sense. You can find more recent statements with these triangles, but nothing that addresses the difficulties that Hayek found, much less anything that surmounts them.

The Centre Of The Solving Subsystem In A Model With Fixed Capital And Scarce Land

1.0 Introduction

This post revisits my example with fixed capital and two types of land. It presents, by means of an example, the concept of the centre of a solving subsystem. Quadrio Curzio & Pellizzari (2010) introduce the solving subsystem in models of rent so as to first solve the price equations without rent. Schefold (1989) introduces the centre of the price system for a pure fixed capital model to, following Sraffa, initially eliminate the prices of old machines from price equations. As far as I know, nobody has combined these concepts before.

The concept of a solving subsystem clarifies how a switch point can lie along a single wage curve. A system of equations for prices is associated with each technique. Each operated process contributes an equation equating revenues and costs. The revenues can include the prices of joint products, and costs include a charge for the rate of profits on advanced capital goods. A last equation specifies the value of the numeraire as unity. In models of extensive rent, a subsystem can be formed from the processes that characterize industrial processes, with no inputs from land, and processes run on land that are not scarce. The resulting subsystem, with the equation for the numeraire concatenated, can be solved, given the rate of profits, for the wage and the prices of produced commodities. In models of intensive rent, the solving subsystem includes the equations for industrial processes and a linear combination of the equations for the processes that operate on one type of land to the limits of its endowment. As Sraffa (1960) explains, a variable for rent is eliminated by this linear combination. In the case of extensive rent, with no joint production otherwise, the solving subsystem also applies to a model of single production. In any case, the solution to the solving subsystem can then be used to find rents. The example in this post, extends the concept of a solving subsystem to a case with extensive rent and fixed capital. I do not know if the concept of a solving subsystem can usefully apply to joint production more generally

The centre of a pure fixed capital system (Schefold 1989) helps solve the price system of a pure fixed capital system. Joint utilization of machines does not exist in any process in a model of pure fixed capital. Old machines are not consumer goods. In the example, a single commodity is a consumption good and acts as numeraire. Old machines may be freely disposed of; no cost arises in junking a machine, including before its technical life is complete. Nice properties of single production systems generalize to such cases of fixed capital. In particular, the "determination of the cost-minimising technique is independent of the structure of requirements for use" (Huang, 2019). The cost-minimizing technique can be determined by the construction of the wage frontier. These properties are not retained in the combination of pure fixed capital with scarce land. The centre still helps solve the price system.

2.0 Technology, Endowments, Final Demand

Tables 1 and 2 specify the technology. This technology extends an example of fixed capital from Baldone (1974). Labor uses circulating capital to manufacture a machine in process I. The machine has a physical life of three years. Labor uses circulating capital and the machine to produce corn on type 1 land in processes II, III, and IV. The machine is operated on type 2 land in processes V, VI, and VII. A process that produces corn jointly produces a machine one year older than the machine used as input, up to its physical life. One hundred acres of each type of land are assumed to exist. Final demand is for 87 bushels corn, a level that ensures one or the other type of land is scarce. The numeraire is a bushel of corn.

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	0	0	0
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

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	0	1	0	0
Type 1 2-Yr. Old Machines	0	0	0	0	0	1	0

3.0 Techniques

Tables 3, 4, and 5 specify the techniques that may be chosen with this technology. Alpha, Beta, and Gamma differ in the economic life of the machine on non-scarce, type 1 land. No processes are operated on type 2 land. Under Delta, Epsilon, and Zeta, on the other hand, type 1 land is not farmed at all, and the economic life of the machine varies among the techniques in the processes operated on type 2 land. The remaining techniques fully cultivate one or the other type of land and require rent to be paid to landlords

Table 3: Techniques of Production with Non-Scarce Land

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

Table 4: Techniques of Production with Type 1 Land Scarce

Technique	Processes	Type 1 Land	Type 2 Land
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

Table 5: Techniques of Production with Type 2 Land Scarce

Technique	Processes	Type 1 Land	Type 2 Land
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

Under techniques Eta through Omicron, type 1 land is fully farmed and pays rent. Under Eta, Theta, and Iota, the machine is operated for only one year on type 2 land and then discarded. The techniques differ on the economic life of the machine on type 1 land. Under Kappa, Lambda, and Mu, the machine is operated for two years on type 2 land, while it is operated for its full physical life of three years under Nu, Xi, and Omicron. Under Pi through Omega, type 2 land is scarce and pays rent. Each technique between Eta and Omicron corresponds to a technique between Pi and Omega in which the same processes are operated. The economic life of the two types of machines are the same in these corresponding techniques. The scale at which the processes are run varies so as to vary which type of land is fully farmed.

4.0 The Price System for Omicron

I consider the price equations for Omicron to illustrate the concepts of the solving subsystem and of the centre. All seven processes are operated under Omicron, and type 1 land is scarce. The following seven displays, in obvious notation, specify the price system for Omicron:

a_1,1(1 + r) + w a_0,1 = p₀

(a_1,2 + p₀)(1 + r) + rho₁ c_1,2 + w a_0,2 = b_1,2 + p_1,1

(a_1,3 + p_1,1)(1 + r) + rho₁ c_1,3 + w a_0,3 = b_1,3 + p_1,2

(a_1,4 + p_1,2)(1 + r) + rho₁ c_1,4 + w a_0,4 = b_1,4

(a_1,5 + p₀)(1 + r) + w a_0,5 = b_1,5 + p_2,1

(a_1,6 + p_2,1)(1 + r) + w a_0,6 = b_1,6 + p_2,2

(a_1,7 + p_2,2)(1 + r) + w a_0,7 = b_1,7

Revenues for operating each process at a unit level are shown on the right-hand side of these equations. Revenues for the first process are obtained by selling new machines. Revenues for the second process result from products of both corn and a type 1 one-year old machine. That type 1 machine, in turn, enters into the advanced costs of the third process, and so on. Type 1 land obtains a rent, and type 2 land is free.

The first equation and the last three of the seven constitute the solving subsystem for Omicron. Given the rate of profits, the solving subsystem specifies the wage, the price of a new machine, and the prices of one-year old and two-year old machines when operated on free type 2 land. The remaining three equations can then be used to find the rent on type 1 land and the prices of one-year old and two-year old machines when operated on type 1 land. The solving subsystem for Omicron is also the solving subsystem for Zeta, Nu, and Xi. In all these techniques, the machine is run for its full physical life of three years on free type 2 land.

The prices of old type 2 machines can be eliminated from the solving subsystem for Omicron. Multiply both sides of the second equation of the solving subsystem by (1 + r)²:

(a_1,5 + p₀)(1 + r)³ + w a_0,5(1 + r)² = b(1 + r)²_1,5 + p_2,1(1 + r)²

Multiply both sides of the third equation of the solving subsystem by (1 + r):

(a_1,6 + p_2,1)(1 + r)² + w a_0,6(1 + r) = b_1,6(1 + r) + p_2,2(1 + r)

Add these two equations and the last equation of the solving subsystem:

where the row vector and matrix in this system of equations is as follows:

The ordered pair consisting of this row vector and matrix is the centre (Schefold 1989) for the solving subsystem for Omicron. Given the rate of profits, this system of matrix equations can be solved for the wage and the price of a new machine. This price system has the form of a price system for a circulating capital model, with the exception of the dependence of the Leontief input matrix and the vector of direct labor coefficients on the rate of profits. Unlike in the model of circulating capital, the wage curve derived from the centre of a pure fixed capital system can slope up for part of its range. The wage frontier of a pure fixed capital system, however, decreases throughout its length (Baldone 1974, Varri 1974).

The prices of old type 1 machines can be similarly eliminated from the full price system for Omicron.

5.0 Conclusion and Questions

The above illustrates the centre of a solving subsystem. In the example, the solving subsystem shows that a system of seven equations for a price system can be decomposed such that a system of four equations is solved first. And the centre of the solving subsystem shows that that system of four equations can be further decomposed so that a system of two equations is solved first.

Perhaps the centre of a solving subsystem can be used to address a theoretical question. Is the wage frontier always decreasing in a model combining fixed capital and rent? Can the wage frontier sometimes slope up?

In a model of extensive rent, the wage frontier is not the outer envelope of the wage curves for the technique. But it is always decreasing. Each wage curve is found from a solving subsystem. And the solving subsystem is from a related circulating capital model. So the wage curves inherit the properties of circulating capital models. The wage frontier is formed from the wage curves of the cost-minimizing techniques and always is decreasing.

In a pure fixed capital model, the wage frontier is the outer envelope of the wage curves for the techniques and is always decreasing. Individual wage curves can be increasing, but the ranges of the rate of profits at which they are increasing is never on the frontier.

I suspect the wage frontier for a model combining extensive rent and fixed capital can be increasing over some range of the rate of profits. This suspicion should be validated by constructing a numerical example. On the other had, if the wage frontier is alwys decreasing in such a model, that should be capable of a proof. And such a proof, if it exists, will probably use the concept of the centre of a solving subsystem.

References

• Baldone, S. (1974), Il capitale fisso nello schema teorico di Piero Sraffa, Studi Economici, XXIV(1): 45-106. Trans. in Pasinetti (1980).
• Huang, B. 2019. Revisiting fixed capital models in the Sraffa framework. Economia Politica 36: 351-371.
• Pasinetti, L.L. 1980. (ed.), 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.
• 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, P. 1974. Prezzi, saggio del profitto e durata del capitale fisso nello schema teorico di Piero Sraffa, Studi Economici, XXIX(1): 5-44. Trans. in Pasinetti (1980).

Factor Demand Curves For An Example With Fixed Capital And Rent

Figure 1: Demand Curve for Labor

I have created and worked through an example in which a machine with a physical life of three years can be used in producing an agricultural commodity on one of two types of land.

My example is one of capital-reversing. It occurs to me that I have not plotted the demand for so-called factors of production in this example. Accordingly, Figure 1 plots the wage against the employment firms want to offer, given final demand. Switch points are horizontal line segments in this graph. Around the 'perverse' switch point, a higher wage is associated with firms wanting to employ more workers.

Given final demand and the rate of profits, a price system is defined for each technique. I can add up the value of the capital goods that must exist at the start of the year to produce the given final demand. Prices of production are used to aggregate heterogeneous goods. Figure 2 shows the demand for capital, in some sense. Here, too, the 'perverse' switch point is indicated for a step function approximation for an increasing demand curve. The value of capital varies between switch points because of price Wicksell effects.

Figure 2: Demand Curve for Capital

A model with both fixed capital and the rent of natural resources is a step towards realism if you want. It is also a step beyond what can be found from empirical Leontief matrices, as I understand it. Still, wages and employment, for example, cannot be explained in the long run by the interactions of well-behaved supply and demand functions in the labor market.

Some Phenomena In Price Theory

I occasionally list theoretical possibilities that I think interesting. Outside of a working paper at Centro Sraffa, I have not managed to publish papers detailing the possibilities listed in this post. Some I have not even written up outside of blog posts. I now know that:

• The recurrence of truncation can occur without the reswitching of techniques. This possibility arises in an example of pure fixed capital, with long-lived machines used in both industries that exist in the example.
• A switch point can lie along a single wage curve, with no other wage curve intersecting at the switch point. This possibility occurs in an example with both fixed capital and rent.
• The order of rentability can be completely opposite the order of efficiency. This possibility can arise in a model that combines extensive and intensive rent.
• The partitioning of parameter spaces by fluke switch points is useful in the analysis of structural economic dynamics with a choice of technique.
• Capital-theoretic paradoxes are transient, in many instances, in secular time (also known as the very long run).

I have some difficulties in writing these up. First, my status as an independent researcher creating examples as a hobby should make reviewers be a bit skeptical. Second, many may not be interested in these refinements. Does not Kurz and Salvadori (1995) provide a definitive statement of post Sraffian price theory? You need to have mastered quite a bit of that to understand the point of any of these. Third, I try to put each in a somewhat more general framework I cast the first, the recurrence of truncation, as an example of the last. I suggest that the second, a switch point along a single wage curve, is an anomalous switch point, a concept I am introducing. I want to say that the third is an example of a special case of a model of intensive and extensive rent in which 'nice' properties of models of extensive rent obtain; wage curves slope down and no issues of the non-existence or multiplicity of cost-minimiing techniques away from switch points arise. Last, when I make such generalizations, I have trouble casting my results into the abstract theorem-proof form needed to be precise.

Is the analysis of structural economic dynamics with a choice of technique an interesting problem? Maybe a book of bookprints never exists at a point of time. Capitalists do not have option of costlessly choosing another page. When a new technique is introduced, it typically dominates the existing technique. On the other hand, I have trouble with part II of Sraffa's book preceding part III. Part II treats joint production, including rent and fixed capital. Part III treats the choice of technique. Which lands to cultivate and what economic lives of machines to adopt are part of the choice of technique. So maybe I should limit my program to aspects of joint production. But I also have some consideration of Harrod-neutral technical progress.

It seems I still have years of work.

An Algorithm Trace For The Truncation Of Fixed Capital

1.0 Introduction

This post revisits my example of the recurrence of truncation without reswitching. In this example, the choice of technique consists of deciding on the economic life of a machine in each industry. I present an application of an algorithm to find the cost-minimizing technique, given the rate of profits. The algorithm needs more elaboration. A trace of the algorithm is a dynamic path through the space of techniques.

2.0 Technology and Techniques

I repeat the parameters that define the example in this section.

Tables 1 and 2 show the inputs and outputs for each process known to the managers of firms. For example, the inputs for the first process, at a unit level of operation, consist of 1/10 person-years, 1/16 bushels corn, and one new machine. The outputs, available after a year, are two new machines and one machine a year older.

Table 1: Inputs for The Technology

Input	Industry
	Machine		Corn
	I	II	III	IV
Labor	1/10	8	43/40	1
Corn	1/16	3/20	1/8	53/200
New Machines	1	0	1	0
One-Year Old Machines (1st type)	0	1	0	0
One-Year Old Machines (2nd type)	0	0	0	1

Table 2: Outputs for The Technology

Output	Industry
	Machine		Corn
	I	II	III	IV
Corn	0	0	1	14/25
New Machines	2	5/2	0	0
One-Year Old Machines (1st type)	1	0	0	0
One-Year Old Machines (2nd type)	0	0	1	0

With this specification of the technology, the economic life of the machine must be chosen in each industry. Table 3 lists the available techniques. The machine is truncated in both industries in the Alpha technique. The machine is operated for its full physical life in both industries in the Delta technique. In Beta and Gamma, the machine is truncated in one industry and operated for its full physical life in the other.

Table 3: Specification of Techniques

Technique	Processes	Notes
Alpha	I, III	Machines truncated in both industries.
Beta	I, II, III	Machines truncated in machine-production.
Gamma	I, III, IV	Machines operated at full physical life in both industries.
Delta	I, II, III, IV	Machines truncated in corn-production.

3.0 An Algorithm for Fixed Capital

I now present a hand-waving, incomplete specification of an algorithm for the choice of technique. This algorithm is supposed to apply when the choice of technique consists exclusively of the choice of the economic life of a machine in various industries.

1. Solve price system, given the rate of profits, for each technique.
2. Identify technique in which machines are operated for two years (longest in example).
• Beta and DELTA in the machine industry
• Gamma and DELTA in the corn industry
3. Find price of old machine in each industry. If it is negative, truncate to longest time in which it is first negative.
4. If a machine is truncated in any industry, repeat previous step.
5. For COST-MINIMIZING technique, prices of old machines are non-negative in all industries.

4.0 Traces

Which order should industries be considered? This is one way the above specification is incomplete. Maybe I should say this is a non-deterministic algorithm. Anyways, Table 4 shows the application of this algorithm starting with the first industry in the example.

Table 4: The Algorithm, Starting with the Machine Industry

Calculate the price of an old machine in the machine industry with Delta prices.
Price negative for 0 ≤ r < 71.2 percent		Price positive for 71.2 percent < r ≤ Rδ
Truncate to Gamma		Keep Delta
Calculate the price of an old machine in the corn industry with Gamma prices.		Calculate the price of an old machine in the corn industry with Delta prices.
Price negative for 0 ≤ r < 70.2 percent	Price positive for 70.2 < r < 71.2 percent	Price positive for 71.2 < r < 87.5 percent	Price negative for 87.5 percent < r < Rδ
Truncate to Alpha	Keep Gamma	Keep Delta	Truncate to Beta
Calculate the price of an old machine in the machine industry with Beta prices.			Calculate the price of an old machine in the machine industry with Beta prices.
Price negative for 0 ≤ r < 70.2 percent			Price positive for 87.5 < r ≤ Rδ
Keep Alpha			Keep Beta

Perhaps the termination criterion for the algorithm should include that a longer economic life of machines has been considered in each industry. In the range of profits in which Alpha is cost-minimizing, I consider extending the economic life of the machine in the corn industry, for the start of the last three rows. These steps extend the algorithm in section 3. Are these steps necessary? When I find a negative price for an old machine in such an extension, can I stop? Or should I, in other examples, continue consider extensions up to the physical life of the machine? I know that truncation can jump from three years, for example, to one year.

Table 5 shows the application of the algorithm starting with the second industry in the example. These two tables illustrate that it does not matter which industry is considered first. I suppose this algorithm, like Christian Bidard's market algorithm, could be distributed across industries, with steps being executed in parallel. I think that if somebody was going to elaborate on this claim, they should consider a specification of market algorithms in a language designed for parallel processing, such as Tony Hoare's Communicating Sequential Processes.

Table 5: The Algorithm, Starting with the Corn Industry

Calculate the price of an old machine in the corn industry with Delta prices.
Price positive for 0 ≤ r < 87.5 percent			Price negative for 87.5 percent < r ≤ Rδ
Keep Delta			Truncate to Beta
Calculate the price of an old machine in the machine industry with Delta prices.			Calculate the price of an old machine in the machine industry with Beta prices.
Price negative for 0 ≤ r < 71.2 percent		Price positive for 71.2 < r < 87.5 percent	Price positive for 87.5 < r ≤ Rδ
Truncate to Gamma		Keep Delta	Keep Beta
Calculate the price of an old machine in the machine industry with Gamma prices.
Price negative for 0 ≤ r < 70.2 percent	Price positive for 70.2 < r < 71.2 percent
Truncate to Alpha	Keep Gamma
Calculate the price of an old machine in the machine industry with Beta prices.
Price negative for 0 ≤ r < 70.2 percent
Keep Alpha

5.0 Conclusion

How should the algorithm be modified for a rate of profits towards the maximum? Can a proof be found that the convergence of the algorithm does not depend on the order in which industries are considered? Once a machine is truncated, is it true, the extension of the economic life a machine need never be considered in any industry?

Rent With Multiple Agricultural Commodities

1.0 Introduction

My next problem might be to explore how to apply an analysis of the orders of efficiency and rentability to a model of rent with multiple agricultural commodities. I would like the possibility of both extensive and intensive rent. This post outlines the structure of a simople numeric example. I have not written down the price systems for each technique. I need to do that to be sure Table 2 is correct.

I should prioritize submitting an article with my most recent model of rent. I should also work through examples in the problems for the approriate chapter in Kurz and Salvadori (1995). As I understand it, nobody has investigated the orders of efficiency and rentability in a model like this. I think Kurz and Salvadori have an existence proof in a special case of joint production, different from the special case I would develop.

Table 1 shows the structure of the technology I am thinking of investigating. Each agricultural commodity can be produced by two processes. The processes differ in which of the two types of land they are operated on, as well as in other coefficients of production.

Table 1: Processes Comprising the Technology

Inputs	Industries
	Iron	Wheat		Rye
	I	II	III	IV	V
Labor	a0,1	a0,2	a0,3	a0,4	a0,5
Type 1 Land	0	c1,2	0	c1,4	0
Type 2 Land	0	0	c2,3	0	c2,5
Iron	a1,1	a1,2	a1,3	a1,4	a1,5
Wheat	a2,1	a2,2	a2,3	a2,4	a2,5
Rye	a3,1	a3,2	a3,3	a3,4	a3,5
OUPUTS	1 ton iron	1 bushel wheat	1 bushel wheat	1 bushel rye	1 bushel rye

Table 3 lists the techniques of production.

• Alpha, Epsilon, and Zeta have the same solving subsystem. Epsilon and Zeta pay extensive rent on type 2 land.
• Delta, Eta, and Theta have the same solving subsystem. Eta and Theta pay extensive rent on type 1 land.
• Iota pays intensive rent on type 1 land. The solving subsystem has joint production.
• Kappa pays intensive rent on type 1 land. The solving subsystem has joint production.
• Lambda pays intensive rent on type 2 land. The solving subsystem has joint production.
• Mu pays intensive rent on type 2 land. The solving subsystem has joint production.
• Nu pays intensive rent on both types of land. The solving subsystem has joint production.

I do not seem to have a technique that pays both extensive and intensive rent. If type 1 land were fully farmed under Epsilon, would the price system not be overdetermined?

Table 2: Technique

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

Anyways, this post presents some thoughts about future research I might explore.

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.

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	0	0	0
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	0	1	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.