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.

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?

Socialism Works in Kerala, India

Socialists and communists have been elected in many places, for significant periods of time. And those places did not necessarily turn into dysopian tyrannies. Often, they did not get further than social democratic policies, improving the lives of most citizens. If I were a member of some of those polities, I would almost certainly have disagreements with details of some policy or other. This post is about one of those places that I do not know much about.

Kerala is a state in India. Currently, the governing party in Kerala is a coalition, the Left Democratic Front (LDF). The Communist Party of India (Marxist) (CPI(M)) is the dominant party. Communists have traded off governing with another coalition, the United Democratic Front, since Kerala's founding, in 1957. They have democratic elections.

The UDF is dominated by the Congress party, which governed India for a long time. I recollect Indira Gandhi, for example. I have no idea how the current national government, headed by Narendra Modi and the Bharatiya Janata Party (BJP), impacts Kerala. My impression is that the BJP raises the question of what is fascism? (By the way, I do not necessarily understand the political positions of some of these newspapers I link to.)

Compared to the rest of India, Kerala has a high Human Development Index (HDI), high literacy, high life expectancy, low poverty rates. For some reason, I have the impression that it has a well-established Information Technology industry.

Bourgeois Propaganda In The Teaching Of Economics

Economists, following a long-exploded, nonsensical theory, have a concept, the 'natural rate of unemployment':

"At any moment of time, there is some level of unemployment which has the property that it is consistent with equilibrium in the structure of real wage rates. At that level of unemployment, real wage rates are tending on the average to rise at a 'normal' secular rate.... A higher level of unemployment is an indication that there is an excess supply of labor that will produce downward pressure on real wage rates. The 'natural rate of unemployment,' in other words, is the level that would be ground out by the Walrasian system of general equilibrium equations, provided there is embedded in them the actual structural characteristics of the labor and commodity markets..." – Milton Friedman (1968), quoted in James K. Galbraith, Time to Ditch the NAIRU

This definition has a number of fatal problems. First, Walras' long run model is overdetermined and inconsistent. An equilibrium cannot be defined with a given endowment of a basket of capital goods, supply matching demand in every market, and the rate of profits (also known as interest) attaining equality in the production of all goods. So that model cannot provide a consistent definition of the natural rate of unemployment.

Second, mayhaps Friedman means to refer to the neo-Walrasian model of intertemporal equilibrium. But that does not work either. Labor services at each point of time are different commodities in that model. You can talk about a time series for unemployment, maybe, but not a natural rate in that model.

Third, empirical evidence shows that the labor market does not work like Friedman imagines. I turn to natural experiments on minimum wages. Andrajit Dube has a number of studies, with co-authors. Dube, Lester, and Reich (2010) is one. Firms do not tend to hire less labor services at higher wages, given the variation found in these experiments. At the macroeconomic level, estimates of the natural rate or of the NAIRU tend to fluctuate, following the actual trend in unemployment. In addition to the previously cited Galbratih article, I also cite Antonella Stirati’s recent Godley-Tobin lecture.

Fourth, the theoretical untenability of explaining wages by demand and supply, as understood in marginal economics, was demonstrated more than half a century ago. Pierangelo Garegnani (1970) is one good exposition.

If economics were a science, students would not have been taught about either the so-called natural rate of unemployment or the NAIRU, for decades. At least, if they were taught so, maybe in a history of thought class, they would also be taught, at least, some of these objections.

I now turn to a web site that purports to provide a tutorial for the student. I find very little about these theoretical and empirical objections in this page. But I do find this:

"To reduce the natural rate of unemployment, we need to implement supply-side policies, such as ...

• ... Making labour markets more flexible, e.g. reducing minimum wages and trade unions.
• Easier to hire and fire workers."

Sometimes the rulers are quite explicit. In 1997, Alan Greenspan extols "heightened job insecurity":

"Atypical restraint on compensation increases has been evident for a few years now and appears to be mainly the consequence of greater worker insecurity. -- Alan Greenspan

So we get right wing political conclusions taught with obsolete theory, as if this were like accepted science, like physics.

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

• Deepankar Basu. A reformulated version of Marx's theory of ground-rent shows that there cannot be any absolute rent. Review of Radical Political Economics 54(4).
• Christian Bidard. 2014. The Ricardian rent theory: an overview. Centro Sraffa working paper 8.
• Christian Bidard. 2018. Ricardo and Ricardians on the order of cultivation. Journal of the History of Economic Thought. 40(3): 389-399.
• A. D'Agata. 1983. The existence and unicity of cost-minimizing systems in intensive rent theory. Metroeconomica.
• Heinz D. Kurz & Neri Salvadori. 1995. Theory of Production: A Long-Period Analysis.
• Alberto Quadrio Curzio. 1980. Rent, income distribution, and orders of efficiency and rentability. In Essays on the Theory of Joint Production (ed. by L. L. Pasinetti).
• Alberto Quadrio Curzio & Fausta Pellizzari. 1999. Rent, Resources, Technology.
• Bertram Schefold. 1989. Mr. Sraffa on Joint Production and Other Essays.
• Piero Sraffa. 1960. Production of Commodities by Means of Commodities. Chapter XI.
• Robert L. Vienneau. 2022. Reswitching in a model of extensive rent. Bulletin of Political Economy 16(2): 133-146.

Elsewhere

• Daniele Girardi. The neoclassical theory of aggregate investment and its criticisms. Review of Political Economy. (working paper)
• Antonella Stirati. 2025. The Godley-Tobin memorial lecture: beyond the NAIRU. Review of Keynesian Economics 13(1): 1-20.
• Graham White. 2010. Labour, Commodities and the labour market: a heterodox perspective. Economic and Labour Relations Review 21(1).
• A November 2025 post on substack, about the Cambridge Capital Controversy, from a Post-Keynesian stude
• An April 2025 post, about the Cambridge Capital Controversy, frm Maura Cassidy Burke, a PhD student at Utrecht University.

Two Dual Linear Programs (LPs) Equivalent To A Linear Complementarity Problem

1.0 Introduction

In a previous post, I mapped a specification of a long run position to a LCP. This specification is in terms of a system of inequalities and equalities, and is in a form appropriate for the application of the direct method to analyze the choice of technique. The LCP supports the application of the Lemke algorithm. Although I have not stepped through the algorithm, I finally understand an aspect of some of Christian Bidard's writings.

This post modifies the LCP so that the matrix M in the LCP has a certain kind of symmetry. With this formulation, the LCP is equivalent to dual LPs.

As far as I know, nobody has written down these dual LPs for analyzing the choice of technique in the special case described by the LCP.

2.0 The Parameters for the Previous LCP

The parameters of a LCP consist of a column vector u and a square matrix M. Where the LCP is equivalent to the specification of a long period position, the column vector is as in Figure 1. The column vector y denotes given final demands for n produced commodities. The row vector a₀ is the direct labor coefficients for each of the m processes comprising the technology.

Figure 1: The Given Vector in the LCP for the Cost-Minimization Problem

The matrix M in the LCP is as in Figure 2. The nxm matrix A is the input matrix. Each column consists of the physical inputs needed to operate a process at unit level. The nxm matrix B is the output matrix. Its columns are the outputs of each process, at a unit level. The scalar r is the given rate of profits.

Figure 2: The Given Matrix in the LCP for the Cost-Minimization Problem

The solution of a LCP consists of two column vectors. In the this case, where the LCP is equivalent to a specification of a long period position, these vectors have a block structure. The components of one solution vector consist of commodity prices and the levels at which each process in the technology is operated in a cost-minimizing solution. A unit of labor is taken as the numeriare.

3.0 A Modification of the LCP

I now consider a variation on the above LCP. Let the matrix M be as in Figure 3. With this modification, the vector y is now total consumption at a point in time along a steady-state growth path. The rate of growth is g.

Figure 3: The Updated Matrix in the LCP for the Cost-Minimization Problem

I make a further assumption that the rate of growth is equal to the rate of profits:

g = r

The matrix M is now skew-symmetric, in which its transpose is equal to its additive inverse. The solution of a LCP in which the matrix parameter has this structure also solves dual LPs.

4.0 Dual LPs

In this case, the primal LP can be written as:

Choose p

To maximize y^T p

Such that:

(1 + r) A^T p + a₀^T ≥ B^T p

p_i ≥ 0, i = 1, 2, ..., n

In other words, prices are set to maximize the value of consumption, while respecting the constraint that the cost of no process, at the given rate of profits, falls below the corresponding revenues.

The dual LP is:

Choose q

To minimize a₀ q

Such that:

[B - (1 + g) A] q ≥ y

q_i ≥ 0, i = 1, 2, ..., m

That is, the levels of operation of the processes comprising the technology are set to minimize total employment, while maintaining the given rate of growth at the point on the steady-state path specified by the consumption basket for the economy. With a unit of labor as numeraire, the objective function of the dual LP can be stated as minimizing total wages.

These dual LPs can be mapped to the LCP with the same mapping as in the previous post, with a couple of modifications. These modifications must be included to include the steady state rate of growth.

5.0 Duality Properties

I want to consider three properties of dual LPs.

If a constraint in the primal LP is met in the solution with an inequality, the corresponding decision variable in the dual LP is zero in the solution. In this context, this duality property is the law of non-operated processes.

If a constraint in the dual LP is met in the solution with an inequality, the corresponding decision variable in the solution to the primal LP is zero. This is the law of free goods.

The values of the objective functions of the dual OPs are equal to one another in the solution. This is Joan Robinson's neo-neoclassical theorem. Given a steady state growth path in which the rate of growth is equal to the rate of profits, the maximum total value of consumption throughout the economy, along that path, is equal to minimum total wages

This trip from the specification of a long period position through an equivalent LCP, the modification of that LCP to have a skew-symmetric matrix, and the consideration of the duality properties of equivalent dual LPs constitutes a novel derivation of Robinson's neo-neoclassical theorem.