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).

Old Papers On Rent And One New One

This post annotates some papers that I want to remind myself of.

Montani (1975) references Quadrio Curzio (in Italian), defines the order of fertility and rentability, notes that they are different, and has something like the reswitching of the order of fertility. He does not have the reswitching of the order of rentability. He treats both extensive and intensive rent, but does not combine them. He notes the wage frontier can slope up under intensive rent. I have to read more closely to see if he already has multiple cost-minimizing techniques. I am under the impression that D'Agata first notice this possibility.

Montet (1979) criticizes Metcalfe and Steedman in that their perversities are more general than they know. Land provides another degree of freedom. They have a wage, rent, rate of profits frontier. I generally do not set equations for natural resources out this way. I once set out an example with heterogeneous labor, relabeling 'land' as 'skilled labor'.

Gibson & McLeod (1983) look at extensive, intensive, and external intensive rent. They go into difficulties of defining basics in joint production. One definition is about the decomposability of matrices and the other is about the rank of some sort of block matrix. They define quasi-basics for the latter. D’Agata has some sort of objection to this. They have interchanges in both the CJE and the RRPE.

Erreygers (1995) considers joint production. Toward the end of his paper, he shows how extensive rent fits into this framework. He wants to avoid setting out another equation in the quantity system to constrain levels of operations of processes from requiring more land to be farmed than exist. And rents should be part of the price vector in the price system, not seperate variables. Kurz & Salvadori (1995) show how to define certain block structured matrices to achieve this end. I think Erreygers may have created this approach.

Ianni (2026) is about international trade, not rent. The theory of intensive and extensive rent can show why most lands are specialized, so the theory may have implications for the theory of international trade. Also, my way of analyzing the choice of technique with long-lasting and given ratios of the rate of profits among industries may have implications for trade. Different countries may be modeled as having different rates of profits.

References

• Erreygers, Guido. 1995. On the uniqueness of cost-minimizing techniques. The Manchester School 63: 145-166
• Gibson, Bill and Darryl McLeod. 1983. Non-produced means of production in Sraffa's system: basics, non-basics and quasi-basics. Cambridge Journal of Economics 7: 141-150.
• Ianni, Guido. 2026. Class struggle and the international division of surplus: The distributive surface in the open economy. Metroeconomica
• Montani, Guido. 1975. Scarce natural resources and income distribution. Metroeconomica 27(1): 68=101.
• Montet, C. 1979. Reswitching and primary input use: a comment. Economic Journal 89 (355): 642-647.

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.

A Switch Point Along The Same Wage Curve With Multiple Agricultural Commodities

Figure 1: Wage Curves Around An Anomalous Switch Points

This post presents another anomalous switch point. A switch point is anomalous in that it has properties that cannot hold for a switch point in a model of single production, with inputs of labor and circulating capital alone.

This example is one with multiple agricultural commodities and intensive and extensive rent. The technology and the endowments of land are the same as in this example.

Required net output, that is, final demand, varies. I start by postulating that final demand consists of 28 bushels wheat and 28 bushels rye. Under this assumption, Alpha, Beta, Epsilon, and Lambda are feasible techniques.

The cost-minimizing technique at a given rate of profits must be:

• Feasible.
• Have non-negative prices for all commodities produced under the technique, have a non-negative wage, and have non-negative rents on all scarce lands.
• Such that no process not operated under the technique obtains extra profits.

Epsilon is cost-minimizing up to a rate of profits of approximately 223.6 percent. A reswitching of the order of efficiency occurs over the range at which Epsilon is cost-minimizing. After the switch point, as illustrated in Figure 1, Alpha is cost-minimizing.

Figure 1 also illustrates a fake switch point at a rate of profits of approximately 219.0 percent. The wage curves for Alpha and Delta intersect at the fake switch point. The wage curve for Alpha is also the wage curves for Epsilon and Zeta. Likewise, the wage curve for Delta is also the wage curves for Eta and Theta. Epsilon is the unique cost-minimizing technique at and around this fake switch point. The prices of produced commodities (iron, wheat, and rye) differ, at the switch point, between the techniques for the two intersecting wage curves. In this sense, the fake switch point resembles the one in the example from Bidard and Klimovsky. The rent on type 2 land is positive under Epsilon, which would not be the case is this switch point was non-fake. Nor are the rents on type 1 land zero under Eta and Theta at this fake.

But consider again the switch point between Epsilon and Alpha. Under Alpha, only type 1 land is farmed, but only partially. Epsilon extends Alpha to produce wheat on type 2 land, to the extent of its endoment. The switch point lies along a single wage curve, which is anomalous.

Suppose that final demand was small enough that both Alpha and Gamma were feasible. For example, Alpha, Beta, Gamma, and Delta are the only feasible techniques when required net output consists of 10 bushels wheat and 10 bushels rye. Then Gamma is cost-minimizing from before the switch point, from approximately 176.8 percent. Alpha is cost-minimizing after the switch point. As with Epslion, under Gamma wheat is produced on type 2 land. But, unlike Epsilon, type 2 land is not farmed under Gamma to the extent of its endowment and the process in which wheat is produced on type 1 land is no longer operated. With this final demand, the example is one of reswitching between Gamma and Delta, at a lower rate of profits than shown.

Or suppose final demand consisted of 30 bushels wheat and 30 bushels rye. Then Beta, Epsilon, Iota, Kappa, and Lambda are feasible. Then this is a switch point between Epsilon and Iota. The same processes are operated under Epsilon and Iota, but which land is scarce varies. Iota is a technique in which landlords obtain intensive rent on type 1 land. The rent on type 1 land is negative under Iota before the switch point.

I have now found switch points:

• Along a single wage curve, with no intersecting wage curve; with variation in which processes are operated between the cost-minimizing techniques.
• In which wage curves intersect, but no variation occurs in which processes are operated between the cost-minimizing techniques.

The above switch point between Epsilon and Alpha combines these two phenomena, in some sense. I have also found fake switch points:

• In which wage curves intersect; the cost-minimizing technique does not vary.

These results suggest that concept of a switch point is not tightly tied to intersections of wage curves in models of joint production.

A Reswitching Example With Extensive And Intensive Rent And Multiple Agricultural Commodities

Figure 1: Wage Curves For Feasible Techniques

1.0 Introduction

This post demonstrates a novel aspect of the reswitching of techniques. The cost-minimizing techniques in the example do not differ in which processes are operated. They differ in which lands are fully cultivated and thus obtain rent. In one technique, two commodities are produced, by distinct processes on the type of land that is fully farmed. In the other, one process, producing one commodity, is operated on the land that pays rent. In other words, the techniques that reswitch pay extensive and intensive rent, respectively.

The reswitching example depends on more than one agricultural commodity being produced. When the technique with extensive rent is cost-minimizing, two processes are operated on type 2 land. Type 2 land is not fully farmed. Two processes producing the same commodity cannot be operated on non-scarce land, away from a switch point, when prices of production prevail.

2.0 Technology, Endowments, Final Demands, and Techniques

Table 1 shows the inputs and outputs for each process known to the managers of firms. Iron is an industrial commodity, produced with no inputs from land. Two types of land are available for producing the agricultural commodities, wheat and rye. Wheat is produced by two processes, each operating on a different type of land. The same is true for rye. Inputs and outputs are specified in physical terms. For example, the inputs for process II, per bushel wheat produced, are 5/2 person-year, the services of one acre of type 1 land, 1/200 ton iron, 1/4 bushels wheat, and 1/300 bushels rye. Each process exhibits constant returns to scale (CRS), up to the limits imposed by the endowments of the lands.

Table 1: Processes Comprising the Technology

Inputs	Industries
	Iron	Wheat		Rye
	I	II	III	IV	V
Labor	a0,1 = 1/3	a0,2 = 5/2	a0,3 = 7/20	a0,4 = 1	a0,5 = 3/2
Type 1 Land	0	c1,2 = 1	0	c1,4 = 2	0
Type 2 Land	0	0	c2,3 = 5	0	c2,5 = 1
Iron	a1,1 = 1/6	a1,2 = 1/200	a1,3 = 1/100	a1,4 = 1	a1,5 = 0
Wheat	a2,1 = 1/200	a2,2 = 1/4	a2,3 = 3/10	a2,4 = 0	a2,5 = 1/4
Rye	a3,1 = 1/300	a3,2 = 1/300	a3,3 = 0	a3,4 = 0	a3,5 = 0
OUPUTS	1 ton iron	1 bushel wheat	1 bushel wheat	1 bushel rye	1 bushel rye

The specification of the problem is completed by defining the available endowments of land and the level and composition of final demand. Accordingly, assume 100 acres of each type of land are available. Suppose the required net output, also known as final demand, consists of 8 bushels wheat and 60 bushels rye.

Table 2 shows the available techniques for these parameters. Land is free for techniques Alpha, Beta, Gamma, and Delta. Techniques Epsilon, Zeta, Eta, and Theta pay extensive rent. Intensive rent is obtained by landlords for Iota, Kappa, Lambda, Mu, and Nu. Under Nu, intensive rent is obtained on both types of land.

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

Consider the Delta technique, for an example of relationships between these techniques. Under Delta, both wheat and rye are grown on type 2 land, but generally not to the limit imposed by the endowments of land. These same wheat and rye-producing processes are operated in Eta and Theta. In both of these techniques with extensive rent, type 2 land is still not farmed to the extent of its endowment. In Eta, wheat is produced on type 1 land to the extent of the endowment of type 1 land. In Theta, rye is produced on type 1 land to the extent of its endowment. But compare Theta to Mu. In Mu, the processes in Delta are also supplemented by growing rye on type 1 land, as in Theta. Type 1 land is not totally farmed, however, under Mu. Type 2 land is totally farmed to the extent of its endowment, with both wheat and rye contining to be produced in parallel on that type of land.

3.0 Quantity Flows and Feasible Techniques

The space of the final demand vector can be partitioned into regions by where each technique is feasible. Assume the final demand for iron is zero. Figure 2 shows a partition of the two-dimensional space for net output of wheat and rye. The point of final demand is indicated for the reswitching example in this post, with a final demand of 8 bushels wheat and 60 bushels rye.

Figure 2: Final Demands for Feasible Techniques

At the lowest level of final demand, Alpha, Beta, Gamma, and Delta are feasible. Land is in excess supply, and the model reduces to a model of circulating capital. As output expands, towards the specified final demand for the example, Delta becomes infeasible. At the limit of Delta's fesibility, net outputs of wheat and rye can be traded off. Lambda and Mu become feasible, with type 2 land obtaining intensive rent.

As final demand continues to expand, Alpha becomes infeasible. Here, too, at the limit for Alpha, the net outputs of wheat and rye can be traded off. Iota and Kappa are the other techniques that pay intensive rent, and they relate to Alpha in the same way that Lambda and Mu relate to Delta. The feasible techniques are now Beta, Gamma, Iota, Kappa, Lambda, and Mu.

As output continues to expand towards the specified point of final demand, Gamma becomes infeasible, with a constraint imposed by the endowment of type 1 land. Iota becomes infeasible, as well. Epsilon and Theta, which pay extensive rent, become feasible. For the reswitching example, Beta, Theta, Kappa, Lambda, and Mu are feasible.

4.0 Price Equations

A system of equations is defined for prices of production for each technique. I take Mu as an example.. Processes I, III, IV, and V are operated under Mu. Processes III and V combine to bring the entire endowment of type 2 land under cultivation. The prices of production for Mu, in obvious notation, satisfy the following equations

(p₁ a_1,1 + p₂ a_2,1 + p₃ a_3,1)(1 + r) + w a_0,1 = p₁

(p₁ a_1,3 + p₂ a_2,3 + p₃ a_3,3)(1 + r) + rho₂ c_2,3 + w a_0,3 = p₂

(p₁ a_1,4 + p₂ a_2,4 + p₃ a_3,4)(1 + r) + w a_0,4 = p₃

(p₁ a_1,5 + p₂ a_2,5 + p₃ a_3,5)(1 + r) + rho₂ c_2,5 + w a_0,5 = p₃

Each equation applies to a process operated under Mu. It is assumed that wages and rents are paid out of the surplus product, at the end of the period of production. The prices of advanced capital goods incur a charge for the rate of prices. A final equation equates the price of the numeraire to unity.

p₁ d₁ + p₂ d₂ + p₃ d₃ = 1

The above equations specify prices of production for Mu. A similar price system characterizes prices of production for each of the other techniques in the example.

The price system for Mu consists of five equations in six variables, r, w, p₁, p₁, p₁, and rho₂. The solution has one degree of freedom. If the rate of profits is taken as given, each of the other five variables can be expressed as a function of the rate of profits. The wage curve is a function of the rate of profits for a given technique. The rent curve is the corresponding function for rent.

The wage curves for the feasible techniques are plotted in Figure 3, at the top of this post. Wage curves are only shown for a technique for the ranges of the rate of profits in which rents are positive. Figure 3 shows graphs of the rent curves for the feasible technique. No rent is paid under Beta, and Beta is never cost-minimizing. Theta is cost-minimizing at high and low rates of profits. Mu is cost-minimizing for intermediate rates of profits.

Figure 3: Rent Curves for Feasible Techniques

5.0 Cost-Minimizing Techniques

Assertions above about the ranges of the rates of profits in which Theta and Mu are cost-minimizing have yet to be demonstrated. The cost-minimizing technique at a given rate of profits is:

• Feasible
• Such that no price of a produced commodity, wage, rate of profits, or rent is negative
• Such that extra profits cannot be obtained, at prices associated with the technique, by operating processes outside the technique

Extra profits exist when the difference between revenues and costs for a process is positive. Costs include a charge on the prices of the advanced capital goods for the given rate of profits, and the difference is taken for a unit level of operation of the process. Prices of production for a technique are such that extra profits are zero for the processes operated by that technique.

Figure 4 illustrates extra profits for Beta. Processes III and IV are not operated by Beta. Extra profits can always be obtained, under Beta, by operating process IV, whatever the rate of profits for which prices are found. Beta is never cost-minimizing.

Figure 4: Beta is Never Cost-Minimizing

Figure 5 shows that Theta is cost-minimizing whenever rent is positive for prices of production for the technique. The graph also makes obvious that the conditions, that rent be non-negative and that extra profits are not available, are independent of one another.

Figure 5: Theta is Cost-Minimizing at High and Low Rates of Profits

Figures 6 and 7 demonstrate that Kappa and Lambda are never cost-minimizing. Figure 8 demonstrates that Mu is cost-minimizing for the range of the rate of profits in which rent is positive for prices of production associated with the technique.

Figure 6: Kappa is Never Cost-Minimizing

Figure 7: Lambda is Never Cost-Minimizing

Figure 8: Mu is Cost-Minimizing at Intermediate Rates of Profits

6.0 The Demand for Labor and for Capital

The analysis of the choice of technique need make no mention of supply and demand functions. But the results allow us to plot the real wage against employment, for the given final demand (Figure 9). The relation can be interpreted as an economy-wide demand curve for labor. The switch points appear as horizontal segments in this graph. The 'perverse' switch point is a step function approximation to an upward-sloping demand curve for labor. The widespread tendency to draw downward-sloping labor demand functions, given ideal assumptions such as competitive markets, no search costs, and so on, lacks a coherent justification.

Figure 9: The Demand for Labor

The rate of profits can also be plotted against the value of advanced capital goods (Figure 10). Here, too, switch points are horizontal segments. The value of capital is the iron, wheat, and rye advanced at the start of the production period and evaluate at prices of production. The variation of the value of capital between switch points is known as a price Wicksell effect. The variation at a switch point across techniques is the real Wicksell effect. What happens at the 'perverse' switch point has also been called reverse capital deepening. At any rate, justification for drawing downward-sloping demand curves for capital, by default, is also lacking.

Figure 10: The Demand for Capital

7.0 Conclusion

This post extends the well-established critique of economic theory to which Sraffa (1960) is a prelude. In this example, the techniques that reswitch do not differ in which processes are operated. They differ in the scale at which these processes are operated, thereby resulting in a variation in which lands are scarce. The usual consequences of 'perverse' switch points appear in which, for example, a higher wage is associated with firms wanting to employ more workers to produce a given final demand.

Innovation Hurting Workers Or Capitalists

Figure 1: The Wage Frontier Is The Inner Envelope Of The Wage Curves For Feasible Techniques

1.0 Introduction

This post presents a solution to the homework problem 7.13 in Kurz & Salvadori (1995), Chapter 10. They assign credit for this problem to Antonio D'Agata. I extend it to include a negligible industrial commodity, as in my outline of a model with rent, multiple lands, and multiple agricultural commodities.

Two agricultural commodities, wheat and rye, are produced by the example economy. If only processes II and V existed, each commodity could only be produced on one type of land. With the given final demand and the given endowments of land, no land would be scarce and no landlords could obtain rent. Suppose innovations introduce new processes, III and IV, so that each commodity could be produced on each type of land. As a result, no long period exists in a range of the rate of profits towards its maximum, and landlords obtain rent at a lower rate of profits.

Nobody else, as far as I know, has considered the orders of efficiency and of rentability in a model with multiple agricultural commodities. Maybe I need to read further in Quadrio Curzio & Pellizzari's book.

2.0 Technology, Endowments, Final Demands, and Techniques

Table 1 shows the inputs and outputs for each process known to the managers of firms. Two types of land are available for producing the agricultural commodities, wheat and rye. Wheat is produced by two processes, each operating on a different type of land. The same is true for rye. Inputs and outputs are specified in physical terms. For example, the inputs for process II, per bushel wheat produced, are one person-year, the services of one acre of type 1 land, a tiny fraction of a ton iron, 3/10 bushels wheat, and 1/10 bushels rye. Each process exhibits constant returns to scale (CRS), up to the limits imposed by the endowments of the lands.

Table 1: Processes Comprising the Technology

Inputs	Industries
	Iron	Wheat		Rye
	I	II	III	IV	V
Labor	a0,1 = 0.0001	a0,2 = 1	a0,3 = 3/2	a0,4 = 1/10	a0,5 = 1/2
Type 1 Land	0	c1,2 = 1	0	c1,4 = 1	0
Type 2 Land	0	0	c2,3 = 5	0	c2,5 = 2
Iron	a1,1 = 0.00001	a1,2 = 0.00001	a1,3 = 0.00001	a1,4 = 0.00001	a1,5 = 0.00001
Wheat	a2,1 = 0.00001	a2,2 = 3/10	a2,3 = 1/10	a2,4 = 1/10	a2,5 = 1/5
Rye	a3,1 = 0.00001	a3,2 = 1/10	a3,3 = 3/10	a3,4 = 1/5	a3,5 = 1/10
OUPUTS	1 ton iron	1 bushel wheat	1 bushel wheat	1 bushel rye	1 bushel rye

The specification of the problem is completed by defining the available endowments of land and the level and composition of final demand. Accordingly, assume 100 acres of each type of land are available. Suppose the required net output, also known as final demand, consists of 15 bushels wheat and 35 bushels rye.

Table 2 shows the available techniques for these parameters. Land is free for techniques Alpha, Beta, Gamma, and Delta. Techniques Epsilon, Zeta, Eta, and Theta pay extensive rent. Intensive rent is obtained by landlords for for Iota, Kappa, Lambda, Mu, and Nu. Under Nu, intensive rent is obtained on both types of land.

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

Suppose only processes I, II, and V are known by managers of firms. Then only the Beta technique is available. Wheat is grown on type 1 land, and rye on type 2 land. The endowments of land provide an upper limit on the level of final demand that can be feasibly satisfied. The innovations that make processes III and IV available provide a choice of technique, including which grains should be grown on which lands. The limits to feasible final demands are increased.

3.0 Quantity Flows

Beta, Kappa, Lambda, and Nu are feasible for the given final demand. Figure 2 shows which techniques are feasible for final demand consisting of any combination of specified quantities of wheat and rye. One type of land is farmed under both Alpha and Delta, with wheat and rye each produced on that type. The maximum final demand for each of these techniques is a downward-sloping straight line. The outer limits for Beta and Gamma have segments where constraints for each type of land kick in.

Figure 2: Feasible Final Demands

Techniques in which extensive rent is paid extend the two techniques with non-scarce land in which both lands are farmed. Epsilon and Theta become feasible when Gamma is no longer feasible. They differ in which type of land, still non-scarce, becomes used to grow both wheat and rye. The other type of land is cultivated to the extent of its endowment. Eventually the non-scarce land, on which both wheat and rye are produced, becomes scarce Zeta and Eta have the same relationship to Beta.

The limits of the final demand for techniques which pay intensive rent also relate to the boundaries on final demand for the other techniques. The maximum final demand for Alpha is the minimum for Iota and Kappa. Iota and Kappa both grow wheat and rye on type 1 land, as in Alpha, but to the full extent of its endowment. They vary with whether non-scarce type 2 land is used to produce wheat or rye. In the same way, the maximum final demand for Delta is the minimum for Lambda and Mu. When the maximum final demand for Iota is the maximum for Gamma (the minimum for Theta), type 2 land is not a constraint. When the maximum for Iota is the maximum for Epsilon (the minimum for Nu), both types of land are constraints. The maximum final demand for Kappa relates to the maximum for Beta and for Zeta in the same way. Likewise, the maximum for Lambda relates to the maximum for Beta and Eta. Finally, the maximum Mu is either the maximum for Gamma or Theta.

4.0 Price Systems

A system of equations for prices is associated with each technique. The going rate of profits is made in each process operated in the technique. I assume wages are paid out of the surplus product at the end of the period required to operate a technique. Rent is also paid on scarce land at the end of the period. A final equation sets the price of the numeraire to unity.

The variables defined by the price system consist of the rate of profits; the wage; the prices of iron, wheat, and rye; and the rents per acre of the two types of land. They are defined up to a degree of freedom. As usual, I take the dependence of the wage on the rate of profits as expressing that degree of freedom. Figure 1, at the top of this post, plots the wage curves for the four feasible technques. Figures 3 plots the rent curves.

Figure 3: Rent Curves

5.0 The Choice of Technique

For a technique to be cost-minimizing at a given rate of profits, the following must be true:

• It must be feasible.
• No price of a commodity, wage, or rent of a type of land can be negative.
• Extra profits cannot be obtained, at the prices associated with the technique, by operating a process not in the technique.

Extra profits are obtained if the difference between revenues and costs for a process, with the going rate of profits charged on advances for capital goods, is positive. Accordingly, I check whether a technique is cost-minimizing by plotting the difference between revenues and costs, for each process, with the prices of that technique.

Process IV pays extra profits throughout the range of the rate of profits in which the price system for Beta has positive orices and a positive wage (Figure 4). Gamma results from replacing the rye-producing process in Beta with process IV. Zeta and Kappa result from producing rye of both types of land. Gamma would be cost-minimizing at a rate of profits greater than approximately 65 percent if it were feasible. But only Kappa, out of Gamma, Zeta, and Kappa, is feasible. Above a rate of profits of approximately 100 percents, processes III and IV both obtain extra profits under Beta. So Beta is not cost-minimizing at any rate of profits.

Figure 4: Extra Profits at Beta Prices

At Kappa prices, process III obtains extra profits at a rate of profits above approximately 18.17 percent. Kappa is cost-minimizing only for rates a profits below this switch point.

Process IV obtains extra profits at Lambda prices for the entire range at which the rent on type 2 land is non-negative for the Lambda price system. Lambda is never cost-minimizing.

All processes are operated under the Nu technique. So extra profits cannot be obtained. But the rent on type 2 land is positive under Nu only when the rate of profits is greater than 18.7 percent. Thus, Nu is not cost-minimizing for a low rate of profits.

The maximum rate of profits for Nu is approximately 153.8 percent. The maximum for Beta is approximately 167.9 percent. Between these limits, Beta is feasible. The wage and the prices of the three produced commodities are positive in the price system for Beta. Both lands are free. Nevertheless, Beta is not cost-minimizing, and a long period position does not exist.

6.0 The Orders of Efficiency and Rentability

The wage curve for the Alpha technique lies on the outer envelope curve, for small rates of profits. The second wage curve, up to a rate of profits of approximately 65 percent, is Gamma's. After that rate of profits, up to the maximum, the wage curve for Gamma is the outermost. These wage curves are not shown in Figure 1.

Only type 1 land is farmed under Alpha. Both types are farmed in Gamma. As final demand expands for a low rate of profits, first type 1 land is cultivated and then both types are farmed. For a larger rate of profits, both types are initially cultivated together. Thus, the order of efficiency varies from type 1, 2 lands to an order in which they are tied (Table 3).

The order from high rent to low rent lands is type 1, 2, whether Kappa or Nu is cost-minimizing. Under Kappa, type 2 land is not scarce and is free. The orders of efficiency and rentability match for low rates of profits, up to a rate of profits in the range in which Nu is cost-minimizing. They differ for higher rates of profits insofar as the order of rentability is not tied.

Table 3: The Choice of Technique

Rate of Profits (Percent)		Technique	Order of Efficiency	Order of Rentability
Minimum	Maximum
0	18.2	Kappa	Type 1, 2	Type 1, 2
18.2	65.1	Nu
65.1	153.8		Type 1 and Type 2 tied

Suppose you take the order of efficiency as showing which lands, at a given rate of profits, contribute most to production. Since the order of rentability can differ from the order of efficiency, prices in competitive markets do not necessarily reward you for the contributions of the factors of production that you own. This conclusion is aside from the doctrine of Henry George.

This example is extremely restricted, when it comes to examining the orders of efficiency. The posibility of ties in the order of efficiency is a new possibility raised by the existence of multiple agricultural commodities. (I suppose you could look for different techniques having identical wage curves in a model of extensive rent and one agricultural comodity.)

7.0 Conclusion

The example shows how innovations create complications in the analysis of long run positions. In the example, innovations lead to the possibility of a class of landlords to come into existence. I doubt this happened like this anywhere. For a certain range of the rate of profits, a long period position no longer exists.

Kurz and Salvadori (1995) have additional numeric examples with multiple agricultural commodities. I should be able to create more. I am interested in examples with the reswitching of the order of fertility, as well as examples in which techniques with extensive and intensive rent are simultaneously feasible.

A Numerical Example Of Reswitching In A Model With Extensive Rent

Figure 1: Wage Curves for Epsilon and Theta

1.0 Introduction

This post presents a numerical example of reswitching. In this example, satisfying requirements for use necessitates farming some type of land to its full extent. That type varies with the rate of profits. Rent is paid on the scarce quality of land.

Two switch points exist in the example. Which type of land is scarce varies at the switch points. Around one switch point, the technique adopted at a higher wage requires less labor, across the entire economy, to produce the required net output. Around the other switch point, the technique adopted at at a higher wage requires more labor. It is a mistake to insist that wages and employment are determined, in the long run, at the intersection of well-behaved supply and demand curves in the labor market.

I do not know of any numeric example elsewhere of reswitching in a model with rent. The possibility of such, however, would not be a surprise to many developers of the theory. Perhaps I have missed something from Schefold or Quadrio Curzio. This example is part of a larger example I have presented earlier.

2.0 Technology, Endowments, Final Demand, and Techniques

Technology consists of three constant-returns-to-scale (CRS) processes (Table 1). Each process is specified by inputs of labor (person-years), services of a type of land (acres), iron (tons), and corn (bushels). This specification also includes the output (ton iron or bushel corn) of each process when operated at a unit level. No land is used in producing the industrial commodity, that is, iron. One process is available to operate on each of the two types of land. Each of these process produces the agricultural commodity, that is, corn. The scale for producing corn is limited by endowments of land.

Table 1: The Coefficients of Production

Input	Industry
	Iron	Corn
	I	II	III
Labor	a0,1 = 1	a0,2 = 9/10	a0,3 = 3/5
Type 1 Land	c1,1 = 0	c1,2 = 1	c1,3 = 0
Type 2 Land	c2,1 = 0	c2,2 = 0	c2,3 = 49/50
Iron	a1,1 = 9/20	a1,2 = 1/40	a1,3 = 3/2000
Corn	a2,1 = 2	a2,2 = 1/10	a2,3 = 9/20

I assume that endowments consist of 100 acres of each type of land. Required net output, also known as final demand, consists entirely of corn, and the required net output is taken as the numeraire. For the example, final demand is assumed to be 125 bushels of corn. At least some of both types of land must be farmed to produced the final demand. (For this property to hold, final demand must be between approximately 80.91 and 136.5 bushels corn.)

The processes defined by the technology can be combined in four techniques (Table 2). In the Alpha and Beta techniques, only one process is operated to produce corn. No land is scarce, and capitalists do not pay rent to landlords. In Epsilon and and Theta, two processes are operated to produce corn. One type of land is fully farmed, and landlords obtain rent on that type of land.

Table 2: Techniques of Production

Technique	Processes	Land
		Type 1	Type 2
Alpha	I, II	Partially farmed	Fallow
Beta	I, III	Fallow	Partially farmed
Epsilon	I, II, III	Partially farmed	Fully Farmed
Theta	I, II, III	Fully Farmed	Partially farmed

3.0 Quantity Flows

Which techniques are feasible varies with required net output. This variation does not arise in models with just circulating capital and no scarce land. Any level and composition of net output can be produced in those models. Also, I want to consider the variation in labor and capital-intensity with the technique. For both these reasons, I need to consider quantity flows.

Table 3 presents quantity flows for a particular level at which the first two processes are operated. The third process is operated at a level of zero. As with the other examples in this section, the total inputs of iron across all processes are replaced by the output of iron produced by the first process. Some corn is left over, as a surplus, after the inputs of corn are replaced by the output of corn from the second process. The table depicts a vertical integration, in which the total input of labor produces the surplus output of corn.

The levels of operation in Table 3 are set such that type 1 land is totally farmed. This combination of flows can be viewed as an extreme case of Alpha, in which no land receives a rent. It is the highest level at which Alpha can be operated, with a surplus output of only corn. For any increase in net output, the third process must also be operated, and type 1 land receives a rent. So these quantity flows can also be viewed as the lowest level at which Theta is operated.

Table 3: Alpha and Theta Quantity Flows at an Extreme

Input	Industry
	Iron	Corn
	I	II	III
Labor	a0,1q1 = 50/11	a0,2q2 = 90	a0,3q3 = 0
Type 1 Land	c1,1q1 = 0	c1,2q2 = 100	c1,3q3 = 0
Type 2 Land	c2,1q1 = 0	c2,2q2 = 0	c2,3q3 = 0
Iron	a1,1q1 = 45/22	a1,2q2 = 5/2	a1,3q3 = 0
Corn	a2,1q1 = 100/11	a2,2q2 = 10	a2,3q3 = 0
OUTPUT	q1 = 50/11	q2 = 100	q3 = 0

Table 4 shows another set of levels at which the three processes can be operated. In this case, type 2 land is totally farmed and obtains a rent. Type 1 land is also farmed, but not completely. Net output in this case, for the Epsilon technique, is the same amount of corn as in the previous example.

Table 4: Epsilon Quantity Flows for the Same Net Ouput

Input	Industry
	Iron	Corn
	I	II	III
Labor	a0,1q1 = 81650/47971	a0,2q2 = 122940/4361	a0,3q3 = 3000/49
Type 1 Land	c1,1q1 = 0	c1,2q2 = 136600/4361	c1,3q3 = 0
Type 2 Land	c2,1q1 = 0	c2,2q2 = 0	c2,3q3 = 100
Iron	a1,1q1 = 73485/95942	a1,2q2 = 3415/4361	a1,3q3 = 15/98
Corn	a2,1q1 = 163300/47971	a2,2q2 = 13660/4361	a2,3q3 = 2250/49
OUTPUT	q1 = 81650/47971	q2 = 136600/4361	q3 = 5000/49

Table 5 shows quantity flows for producing the largest possible surplus product of corn. More can only be produced with the discovery of more land or technical innovation that reduces at least some coefficients of production. Both types of land are fully farmed to the extent of their endowments.

Table 5: Epsilon and Theta Quantity Flows at an Extreme

Input	Industry
	Iron	Corn
	I	II	III
Labor	a0,1q1 = 2600/539	a0,2q2 = 90	a0,3q3 = 3000/49
Type 1 Land	c1,1q1 = 0	c1,2q2 = 100	c1,3q3 = 0
Type 2 Land	c2,1q1 = 0	c2,2q2 = 0	c2,3q3 = 100
Iron	a1,1q1 = 1170/539	a1,2q2 = 5/2	a1,3q3 = 15/98
Corn	a2,1q1 = 5200/539	a2,2q2 = 10	a2,3q3 = 2250/49
OUTPUT	q1 = 2600/539	q2 = 100	q3 = 5000/49

Table 6 summarizes results from the above tables. The same net output is produced, by the Epsilon and Theta techniques, in the first two rows. The same is true of the last two rows. This excursion into the analysis of quantity flows demonstrates the range of required net output, when it consists solely of corn, in which rent is paid in all feasible techniques. Theta is more labor-intensive within this range. (At the upper limit, quantity flows for Epsilon and Theta are identical.)

Table 6: Summary of Quantity Flows

Technique	Net Output (Bushels)	Labor (Person-Yrs.)	Labor Intensity (Person-Yrs. per Bushel)
Epsilon	890/11 ≈ 80.91	4370990/47971 ≈ 91.1	437099/388129 ≈ 1.13
Theta	890/11 ≈ 80.91	1040/11 ≈ 94.55	104/89 ≈ 1.17
Epsilon	73560/539 ≈ 136.5	84110/539 ≈ 156.0	8411/735 ≈ 1.14
Theta	73560/539 ≈ 136.5	84110/539 ≈ 156.0	8411/735 ≈ 1.14

4.0 Prices of Production

A system of equations is available for the prices of production defined by each technique. For an example, I specify the price system for Epsilon. The going rate of profits is made in operating the first process:

(a_1,1 p₁ + a_2,1 p₂)(1 + r) + w a_0,1 = p₁

In this equation, p₁ is the price of iron, p₂ is the price of corn, w is the wage, and r is the rate of profits. The going rate of profits is also made in operating the second process:

(a_1,2 p₁ + a_2,2 p₂)(1 + r) + w a_0,2 = p₂

Type 1 land is not scarce under Epsilon and receives no rent. Thus, rent does not appear in the above equation. The going rate of profits is also obtained in operating the third process:

(a_1,3 p₁ + a_2,3 p₂)(1 + r) + rho₂ c_2,3 + w a_0,3 = p₂

Type 2 land receives a rent, and rho₂ denotes rent-per-acre on type 2 land. Finally, the numeraire has a price of unity:

p₂ d₂ = 1

In this equation, d₂ denotes the amount of corn in required net output in the example, that is, 125 bushels.

The above four equations determine five variables. Thus, the solution has one degree of freedom. I take the wage as a function of the rate of profits in the solution. That is, the wage curve showing a trade-off betwen the proportion of the net output paid to labor and the rate of profits depicts this degree of freedom. Prices, including the rent on type 2 land, are also functions of the rate of profits.

The first two equations in the price system also apply for the price system for Alpha. Along with the equation specifying the numeraire, they provide the 'solving subsystem' for Alpha and Epsilon. The wage curves for Alpha and Epsilon are identical. The third equation in the price system for Epsilon can be solved for rent, given the solution for Epsilon's solving subsystem.

5.0 Choice of Technique

For the given required net output, the Alpha and Beta techniques are not feasible. Epsilon and Theta are feasible. Figure 1, at the top of this post, graphs wage curves, from the price systems for the techniques. Alpha and Epsilon have the same wage curves. Likewise, Beta and Theta have the same wage curves. The wage frontier for the cost-minimizing technique, is, in this example, formed from the inner envelope of wage curves. Rent on scarce land must be non-negative for the technique that is cost-minimizing at a given rate of profits.

Epsilon is cost-minimizing at extreme ranges of the rate of profits. Theta is cost-minimizing in the middle. So this example is indeed one of reswitching. Two switch points exist for this reswitching example. A switch point is labeled as 'perverse' only because phenomena around that switch point contradict obsolete marginalist concepts. Figure 2 shows rent curves. Landlords obtain rent on type 2 land when Epsilon is cost-minimizing and on type 1 land when Theta is cost-minimizing.

Figure 2: Rent Curves for Epsilon and Theta

The analysis of the choice of technique allows you to plot the wage against the labor demanded by firms, given the specified level of net output (Figure 3). This plot may be interpreted as an economy-wide demand curve for labor. Switch points correspond to the horizontal segments on this graph. The ‘perverse’ switch point can be viewed as a step function approximation to an upward-sloping labor demand curve, if you insist on pretending that wages and employment are determined by supply and demand in competitive markets.

Figure 3: Employment as a Function of the Wage

You can also plot the value of advanced capital goods against the rate of profits. These capital goods consist of iron and corn in the example. The plot in Figure 4 can be interpreted as a demand curve for capital. Switch points correspond to the horizontal segments on this graph too. The value of capital goods varies between switch points because of price Wicksell effects. Prices of production generally vary with the rate of profits, given the technique. The 'perverse' switch point here too can be seen as a step function for an upward-sloping demand curve.

Figure 4: The Value of Capital as a Function of the Rate of Profits

6.0 Conclusion

The choice of technique is trivial in this example. Other than at switch points, the cost-minimizing technique:

• Is feasible. The technique can be used to produce the given final demand.
• Pays a positive rent in the solution to the price system for the technique.

Because of the simple structure of the example, only one technique satisfies these conditions, except at switch points, at any given rate of profits less than the maximum.

Reswitching and capital-reversing are compatible with models of land-like, scarce natural resources. Why do so many economists teach theoretically and empirically unfounded models with factor prices determined by interaction of well-behaved supply and demand curves?

References

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