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.

On The Failure Of So-Called Neoclassical Economics

I want to contrast the theories of classical political economists and marginalists up to, say, the 1920s. I take David Ricardo as representative of classical political economy. For purposes of this post, I consider Karl Marx to also be a classical political economist.

For marginalists, I think of Eugen Bohm Bawerk, John Bates Clark, William Stanley Jevons, Alfred Marshall, Leon Walras, Knut Wicksell, and Philip Wicksteed among a host of others. Obviously, I am, at this level of abstraction, ignoring differences among both groups.

Modern economists have established that the classical political economists were broadly correct. And that the marginalists around the time of their intellectual revolution were ultimately incorrect.

Both groups tried to explain roughly the same object with their theories. That is, they proposed theories of long run equilibrium. (Some argue that, like other technical terms used by marginalists, applying the term 'equilibrium' to David Ricardo's theories is not quite correct.) Prices that exist in markets at any time vary. Even the same commodity may be sold at different prices by different buyers and sellers that are located nearby in time and space. Both goups thought, even so, that some sort of center of gravity was attracting these market prices, that they were fluctuating about this center. Anyways, they developed theories about this position. And in these theories, the law of one price would prevail. In competitive markets, the same rate of profits would prevail in all markets.

They did not theorize that a long run equilibrium would ever be reached. Walras, for example, compared his equilibrium to the flat surface of a lake that was always being disturbed by winds and waves.

But the groups differed on what data they took as given in that part of their theories that explained equilibrium prices. For the classical political economists, the givens in this part of the theory consist of:

• Technology
• The real wage
• How much of each commodity is produced.

As a matter of mathematics, these givens are sufficient to explain the prices prevailing in a long run position.

The marginalists have another set of data. These givens consist of:

• Technology
• Tastes
• The endowments of land, labor, and capital, including the initial distribution of these givens among the agents in the model.

As a matter of mathematics, a consistent model of a long run equilibrium cannot be constructed with these givens How to take the endowment of capital is one of those matters that differed among the marginalists. All of their approaches were incoherent.

This post merely echos conclusions that academics came to about half a century ago and have been repeating. I think of Leontief's input-output analysis and of some applications of mathematical programming as empirical work building on a renewed classical political economy.