A Non-Reswitching Theorem Inapplicable To Non-Competitive Markets?

Consider a circulating capital model of the production of commodities. A non-reswitching theorem exists:

Theorem: Suppose a commodity exists which is a basic commodity in all techniques. And a smooth, continuously differentiable production function exists for producing that commodity. Then the reswitching of techniques cannot arise.

Marglin (1984: 285-286) states a theorem like this in which continuously differentiable production functions exist for all commodities. He also states:

"Once again, a result I thought to be original turned out not to have been. Reviewing the literature for these notes, I found a proof of the impossibility of reswitching in a continuous-substitution framework in Burmeister and Dobell (1970)." – Marglin (1984: 542).

Marglin's proof is one by contradiction. Capital-reversing is still possible under the assumption of these theorems.

Pasinetti and Scazzieri (2008) find the theorem in Bruno, Burmeister, and Sheshinski (1966), which I do not recall. They, in turn, attribute the theorem to Martin Weitzman and Robert Solow. Pasinetti and Scazzieri doubt the validity of the theorem.

"It is worth noting that Weitzman–Solow's theorem is simply a consequence of the idea that, in the case of a commodity produced by a neoclassical production function, each set of input–output coefficients ought to be associated in equilibrium with a one-to-one correspondence between marginal productivity ratios and input price ratios. No ratio between marginal productivities would be associated with more than one set of input prices, and this is taken to exclude the possibility that the same technique be chosen at alternative rates of interest, and thus at different price systems. The Weitzman–Solow theorem is at the origin of a line of arguments that has been followed up by a number of other authors, such as David Starrett (1969) and Joseph Stiglitz (1973). These authors have pursued the idea that 'enough' substitutability, by ensuring the smoothness of the production function, is sufficient to exclude reswitching of technique. However, non-reswitching theorems of this type involve that, for each technique of production, the capital stock may be measured either in physical terms or at given prices. For in a model with heterogeneous capital goods, if we allow prices to vary when the rate of interest or the unit wage are changed, there is no reason why the same physical set of input–output coefficients might not be associated with different price systems: even in the case of a continuously differentiable production function, the marginal product of 'social' capital cannot be a purely real magnitude independent of prices. Once it is admitted that 'in general marginal products are in terms of net value at constant prices, and hence may well depend upon what those prices happen to be' (Bliss, 1975, p. 195), it is natural to allow for different marginal productivities of the same capital stock at different price systems. It would thus appear that reswitching of technique does not carry with it any logical contradiction even in the case of a smoothly differentiable production function." Pasinetti and Scazzieri (2008)

I do not know about that. But I have never been clear on how substitutability is supposed to justify marginalist theory.

Suppose rates of profits differ among industries. And that the ratios of rates of profits among industries are stable in the long run. I have shown that the arguments of the Cambridge capital controversy extend to such non-competitive markets. In my paper, I had a reswitching example, in a discrete technology, that did not arise in competitive markets.

I conjecture that the non-reswitching theorem for continuous-substitution does not apply to non-competitive markets.

References

• Bruno, M., Burmeister, E. and Sheshinski, E. 1966. The nature and implications of the reswitching of techniques. Quarterly Journal of Economics 80, 526–53.
• Burmeister, Edwin and A. Rodney Dobell. 1970. Mathematical Theories of Economic Growth. New York: Macmillan.
• Marglin, S. A. 1984. Growth, Distribution, and Prices. Harvard University Press.
• Pasinetti, L. L. and Roberto Scazzieri, R. 2008. Capital theory (paradoxes). The New Palgrave.
• Starrett, D. 1969. Switching and reswitching in a general production model. Quarterly Journal of Economics 83, 673–87.
• Stiglitz, J. 1973. The badly behaved economy with the well-behaved production function. In Models of Economic Growth, ed. J.A. Mirrlees and N.H. Stern. London: Macmillan.

An Example With Intensive And Extensive Rent

Figure 1: Detail on Variation of Rent per Acre with Rate of Profits

1.0 Introduction

This post is the start of an attempt to develop an interesting example with both intensive and extensive rent. A feasible technique exists in the example with both intensive and extensive rent. Yet, it is never cost-minimizing. So this example does not do what I want. I have previously thought about other examples.

The example is an extension of my example of the reswitching of the order of rentability. Such reswitching occurs in this example. But the first switch point of the order of rentability is off the frontier.

I think some perturbation of this example will get me an example where a technique with both intensive and extensive rent is cost-minimizing for some range of the rate of profits. That example will illustrate that the orders of efficiency and of rentability can be analyzed in the context of intensive rent. And these orders need not co-incide in the case of intensive rent too.

2.0 Technology, Resources, Final Demand, and Feasibility

Table 1 presents coefficients of production for the example. Two commodities are produced, iron and corn. Aside from the use of land, joint production is not possible. Multiple types of land (that is, three types) exist. Only one agricultural commodity, corn, can be produced on the processes in which land is used. For one type of land, more than one process can be operated on land. Only one process is known for producing iron, the industrial commodity. Each column in Table specifies the person-years of labor, acres of a type of land, tons of iron, and bushels of corn needed to produce a unit output of the specified commodity.

Table 1: The Coefficients of Production

Input	Industry
	Iron	Corn
	I	II	III	IV	V
Labor	1	0.517	91/250	0.67	3/10
Type I Land	0	0.49	0	0	0
Type II Land	0	0	0.59	0	0
Type II Land	0	0	0	9/20	3
Iron	9/20	0.03744	0.0009	0.067	0.08
Corn	2	0.048	0.27	0.15	0.15

I can define various techniques (Table 2) with this technology. I need all twenty-four letters in the Greek alphabet to specify the techniques. Not all techniques are feasible, given technology, endowments, and requirements for use. Land is not scarce for the Alpha, Beta, Gamma, and Delta techniques, and ownership of land obtains no rent. The Epsilon through Upsilon techniques are examples of extensive rent. One type of land obtains a rent in the Epsilon through Xi techniques. All three types are farmed in Omnicro through Upsilon, and two types obtain a rent. Phi is an example of intensive rent. Chi, Psi, and Omega are examples of the combination of intensive and extensive rent.

Table 2: Techniques of Production

Technique	Processes	Land
		Type 1	Type 2	Type 3
Alpha	I, II	Partially farmed	Fallow	Fallow
Beta	I, III	Fallow	Partially farmed	Fallow
Gamma	I, IV	Fallow	Fallow	Partially farmed
Delta	I, V	Fallow	Fallow	Partially farmed
Epsilon	I, II, III	Partially farmed	Fully Farmed	Fallow
Zeta	I, II, IV	Partially farmed	Fallow	Fully Farmed
Eta	I, II, V	Partially farmed	Fallow	Fully Farmed
Theta	I, II, III	Fully Farmed	Partially farmed	Fallow
Iota	I, III, IV	Fallow	Partially farmed	Fully Farmed
Kappa	I, III, V	Fallow	Partially farmed	Fully Farmed
Lambda	I, II, IV	Fully Farmed	Fallow	Partially farmed
Mu	I, III, IV	Fallow	Fully Farmed	Partially farmed
Nu	I, II, V	Fully Farmed	Fallow	Partially farmed
Xi	I, III, V	Fallow	Fully Farmed	Partially farmed
Omnicron	I, II, III, IV	Partially farmed	Fully Farmed	Fully Farmed
Pi	I, II, III, V	Partially farmed	Fully Farmed	Fully Farmed
Rho	I, II, III, IV	Fully Farmed	Partially farmed	Fully Farmed
Sigma	I, II, III, V	Fully Farmed	Partially farmed	Fully Farmed
Tau	I, II, III, IV	Fully Farmed	Fully Farmed	Partially farmed
Upsilon	I, II, III, V	Fully Farmed	Fully Farmed	Partially farmed
Phi	I, IV, V	Fallow	Fallow	Fully Farmed
Chi	I, III, IV, V	Fallow	Fully Farmed	Fully Farmed
Psi	I, II, IV, V	Fully Farmed	Fallow	Fully Farmed
Omega	I, II, III, IV, V	Fully Farmed	Fully Farmed	Fully Farmed

I assume that 100 acres of each of the three types of land are available. Net output consists of 60 tons iron and 80 bushels corn. This completes the specification of the example.

Under these assumptions, Zeta, Lambda, Omnicron, Pi, Sigma, Tau, Upsilon, and Psi are feasible. Types 1 and 3 land are farmed under Zeta, with process IV being operated on Type 3 land. Which of Types 1 and 3 obtain a rent depends on which land is fully farmed. I should say more here.

3.0 Prices of Production

A system of equations specify prices of production for each technique. All operated processes pay the same rate of profits. Rents and wages are paid out of the surplus at the end of the year. A type of land that is only partially farmed is not scarce and pays no rent. I take the net output as the numeraire.

One degree of freedom exists for the system of equations for each technique. Figure 2, below, shows how the wage varies with the rate of profits for each technique. Figure 3 shows the variation in rent per acre with the rate of profits. Figure 1, at the top of this post, is a detail for an interesting part of Figure 3.

Figure 2: Wage Curves for Feasible Techniques

Figure 3: Rent Curves for Feasible Techniques

4.0 Choice of Technique

A technique is not cost-minimizing if it requires a negative rent to be paid. Rent is negative, under Sigma, for both Type 1 and Type 3 lands. Under Zeta, Omnicron, and Pi, rent on Type 3 land is negtive.

This leaves Lambda, Tau, Upsilon, and Psi as feasible techniques that pay positive rents on scarce lands in some range of the rate of profits. Table 1 lists the cost-minimizing techniques, Upsilon and Tau, in order of an increasing rate of profits.

Table 3: Cost-Minimizing Technique

Range	Technique	Order of Efficiency	Order of Rentability
0 ≤ r ≤ 28.49 %	Upsilon	Type 2, 1, 3	Type 2, 1, 3
28.49 ≤ r ≤ 29.05 %			Type 1, 2, 3
29.05 ≤ r ≤ 35.50 %	Tau
35.05 ≤ r ≤ 43.76 %			Type 2, 1, 3

The order of efficiency, at a given rate of profits, is the order in which different types of land would be brought under cultivation as final demand was increased. This order can be read off of Figure 2 by working downward over the wage curves. Since the wage curves for Sigma and for Zeta, Omnicron, and Pi do not intersect, the order of efficiency does not vary, with the rate of profits, in this example. Type 2 land is partially farmed under Sigma. So Type 2 land is first in the order of efficiency. Type 1 land is partially farmed in Zeta, Omnicron, and Pi. Hence, Type 1 land is next in the order of efficiency for techniques in which all three lands are farmed.

The order of rentability is read off of Figures 1 and 3. The order in which rent per acre decreases varies with the rate of profits. For order of rentability differs from the order of efficiency for rates of profits around the switch point between Upsilon and Tau. A change in the order of rentability occurs around the second intersection between the two rent curves for Tau. This effect is a manifestation of the reswitching of the order of rentability. But the order of rentability varies around any intersection of these curves.

It remains to demonstrate that the above claims about which is the cost-minimizing technique at each rate of profits. Figure 4 shows that Lambda is never cost minimizing. Extra profits can always be obtained at Lambda prices by farming Type 2 land with process III. At low rates of profits, extra profits can also be obtained by farming Type 3 land with the other corn-producing process available for that type of land.

Figure 4: Extra Profits at Lambda Prices

Figures 5 and 6 show that Upsilon is cost-minimizing below the switch point, and that Tau is cost-minimizing at higher rates of profits. In these ranges, extra profits are not available by operating the process not in the technique.

Figure 5: Extra Profits at Upsilon Prices

Figure 6: Extra Profits at Tau Prices

Both intensive and extensive rent are paid when Psi is adopted. Figure 7 demonstrates that Psi is never cost-minimizing. Extra profits are always available, whatever the rate of profits.

Figure 7: Extra Profits at Psi Prices

5.0 Conclusion

This post has illustrated the analysis of the choice of technique in an example with both intensive and extensive rent. Constructing the wage curve is not necessarily the correct method of analysis in models with general joint production. Looking at whether or not extra profits are available for the prices associated with a technique is always applicable.

Elsewhere

• I find Matt McManus has argued in Damage magazine for liberal socialism.
• The liberal socialism developed by Nello and Carlo Rosselli was sort of endorsed by Mussolini. He had them murdered.
• When it comes to Italian exiles from fascism, I could learn more about Pietro Nenni.
• This Ha-Joon Chan article in the Financial Times is behind a paywall and is generating a buzz. "Economics teaching has become the Aeroflot of ideas". But it is available here.

Revolutionary And Reformist Socialists Splitting Around 1900

The second international was the major organization for socialist parties around 1900. The first international had collapsed with struggles for leadership between Marx and anarchists. The German Social Democratic Party, headed by Karl Kautsky, seemed to be the most successful socialist party in the second international. After Engels, Kautsky became the literary executor for Marx. He edited and put out volume 4 of Capital, that is, Theories of Surplus Value.

Eduard Bernstein was Engels' literary executor and therefore a prominent member of the German Social Democratic Party. He had been the editor of Der Sozialdemokrat, the party's newspaper. Perhaps Bernstein was influenced by his acquaintance with members of the Fabian society when he was in exile in London. He looked at the growing wealth of the German workers; the apparent strength of working-class organizations, such as unions; and the SDP representation in the Reichstag. Economic development was not concentrating wealth in a smaller and smaller capitalist class. These trends did not seem to him consistent with the revolution that Marx foresaw. And he said so.

The Preconditions of Socialism and the Tasks of Social Democracy (Die Voraussetzungen des Sozialismus) is the major statement of Bernstein's views, and was published in 1899. It started as articles in Die Neue Ziet, the SDP paper for more theoretical work. Bernstein was called a revisionist, which has ever after been a pejorative among more radical socialists. His theses were argued about at the SDPss Stuttgart Conference, held in October 1898.

Bernstein argued for legislation and peaceful reform in favor of the workers. Socialists should be a parliamentary party. They should agitate for universal suffrage. They should leave businesses, for the most part, in private hands. Socialists should support the development of civil society.

Rosa Luxemburg saw an opportunity to raise her stature in the German SDP. She had already participated in the Stuttgart conference. Others, including Kautsky, also argued against Bernstein. Luxemburg's Reform or Revolution is a classic statement of the radical, anti-revisionist view. She argued against idealism, against petty bourgeois moralism, and against opportunism. Idealism, in this sense, means basing political views purely on intellectual arguments. Emphasizing universal citizenship loses a working-class standpoint. According to Luxemburg, capitalism will inevitably break down. Socialism is a scientific standpoint, given its historical necessity. I do not know that this is in this pamphlet, but Luxemburg famously said that our choice is socialism or barbarianism.

This controversy was echoed in other countries. In France, Jean Jaurès led the reformists. I think of Georges Sorel as an intellectual leader of the revolutionaries. His 1908 book, Reflections On Violence does not strike me as particularly Marxist. You maybe should read 'violence' in the title as what is today called direct action. Sorel, at the time, was an advocate of syndicalism. He was kind of mystical in his emphasis on non-rational motives for mass movements. Hence, his myth of the general strike.

In Italy, Filippo Turati, a founder of the Italian socialist party (PSI), was a reformist. The radicals were called maximalists. They seem to me more positivist than Marx would ever be. They saw the revolution as inevitable, not something to be brought about by political action in the here-and-now. Socialists should organize and educate, holding themselves back until the revolution comes.

In Russia, the split was between the Mensheviks and the Bolsheviks. Apparently, these names mean 'the minority' and 'the majority'. Bolsheviks were only the majority because some supporters of the Mensheviks had walked out of the Second Congress of the RSDLP (Russian Social Democratic Labor Party), at which the split occurred. Julius Martov was an important leader of the moderates, while Lenin headed the Bolsheviks. Lenin's pamphlets, What is to be done? (1902) and Two steps forward, one step back (1904) are essential primary sources here.

Lenin argued for agitation on all fronts, not just for economic improvements for the workers. He wanted an organization of professional revolutionaries, and developed the idea of democratic centralism. The Bolsheviks should have the freest discussion in deciding on policy and tactics. But once a vote has decided the issue, the comrades follow the party line. The establishment of an all-Russian newspaper, Iskra, is the immediate implementation called for in What is to be done?

I ought to say something about Austria.

I have said nothing about Sweden, Denmark, Norway, or Finland. Bernstein provided the intellectual structure for what became democratic socialism and social democracy. I do not know historical details about Scandinavia. He said, "The final goal, no matter what it is, is nothing; the movement is everything."

Some Results From My Research

I have written about my research program before. Here I write about results.

1. A numerical example of a new capital-theoretic paradox, the reswitching of the order of rentabily.
2. A numeric example of, perhaps, another new capital-theoretic paradox, the recurrence of truncation without reswitching or capital-reversing. (I have not yet had this published.)
3. A demonstration that the critique developed in the Cambridge capital controversy extends to non-competitive markets.
4. A diagram illustrating how the analysis of the choice of technique varies with the perturbation of a parameter in models of the production of commodities.
5. A claim that certain capital-theoretic paradoxes are (usually?) only transient in the very long-run, secular time.
6. The discovery and presentation of certain generic structures in parameter spaces, associated with qualitative change in the analysis of the choice of technique. This is the broadest claim in my series of papers.

I do not imagine that Christian Bidard, Heinz Kurz, Neri Salvadori, Bertram Schefold, or Ian Steedman, for example, would be surprised by the first two or three bullet points. I concentrate on producing concrete numerical illustrations. I do not expect ever to state the fifth and sixth point with full mathematical rigor. I know the fifth point cannot be completely general.

Richest Man In The World Kills Hundreds Of Thousands, Will Kill Millions

This post is current events.

I refer to the illegal and unconstitutional elimination of United States Agency for International Development (USAID). This agency was eliminated by the unelected, nazi-saluting Elon Musk, working for the traitor Donald Trump.

Some documentation:

• Cavalcanti et al., 30 June 2025. Evaluating the impact of two decades of USAID interventions and projecting the effects of defunding on mortality up to 2030. The Lancelet.
• Malone. 31 May 2025. Bono criticizes USAid cuts in lengthy interview with podcaster Joe Rogan. Irish Times.
• Ensor. 30 May 2025. DOGE cuts to USAID blamed for 300,000 deaths ' most of them children. The Times.
• Donati et al. 16 May 2025. US aid cuts leave food for millions mouldering in storage. Reuters.
• Belz. 8 May 2025. Aid cuts could disrupt historic drops in child and maternal mortality. Christianity Today.
• Colarossi. 30 March 2025. "It's unacceptable": BU mathematician tracks how many deaths may result from USAID, Medicaid cuts. The Brink.

Still More On Recurrence Of Truncation Without Reswitching

Figure 1: An Example of Structural Economic Dynamics

This post presents a diagram illustrating the effects of a particular kind of technical progress in a model with fixed capital. The example is one with two industries. Machines are produced in the machine industry and used in both the machine and corn industry. All production processes take a year to complete. Machines have a physical lifetime of two years. Managers of firms have a choice in each industry. The economic life of a machine can be one or two years in either industry.

This previous post specifies the technology at time zero. Coefficients of production for inputs and outputs are shown in Tables 1 and 2 in that post. Table 3 defines the four techniques, Alpha, Beta, Gamma, and Delta. Table 1 repeats that specification in a somewhat different format.

Table 1: Specification of Techniques

Technique	Economic Lifetime of Machines
	Machine Industry	Corn Industry
Alpha	One Year	One Year
Beta	Two Years	One Year
Gamma	One Year	Two Years
Delta	Two Years	Two Years

I consider perturbations of coefficients of production that define the amount of circulating capital needed to operate new and old machines, both in producing more new machines and in producing corn. This previous post defines the regions illustrated in the diagram at the top of this post. I repeat the definitions of the regions in Table 2 here. The cost-minimizing techniques, in each region, are listed in the order of an increasing rate of profits or decreasing wage.

Table 2: Overview of Regions

Region	Techniques	Notes
1	Alpha	No switch point.
2	Alpha, Gamma	Lower rate of profits associated with truncation in corn industry, greater output per worker.
3	Alpha, Gamma, Delta	Lower rate of profits associated with truncation, greater output per worker.
4	Alpha, Gamma, Delta, Beta	Recurrence of truncation in corn industry.
5	Alpha, Beta	Lower rate of profits associated with truncation in machine industry, greater output per worker.

I also consider the perturbations of relative markups, yielding a figure much like that at the top of this post. I conclude that both technical progress and changes in market power can have similar effects, in the large. In this example, both can bring about or eliminate the recurrence of the truncation of the economic life of the machine in one industry. This post further illustrates this claim about technical progress.

I consider further perturbations of coefficients of production in this post and this post. Although four coefficients of production decline with technical progress in Figure 1, I have not managed to specify a path through the additional regions in parameter space found in these later posts.

Sewer Socialism Worked In Milwaukee, Wisconsin, In The First Half Of The 20th Century

America's Socialist Experiment

Socialists have been elected mayor, to city councils, and to county commissions in the USA. Emil Seidel was the first socialist mayor in the USA. He was elected mayor of Milwaukee in 1910. Other socialists were elected to the city council. Another socialist, Victor Berger, was elected to the House of Representatives, to represent Milwaukee in Washington, DC.

By concentrating on Milwaukee, I am downplaying the extent to which socialists were elected mayor in cities in the USA. Wikipedia has a list, and so does the University of Washington. More than 130 mayors were socialists. Maybe I want to read David R. Berman's 2022 book, Socialist Mayors in the United States: Governing in an Era of Municipal Reform, 1900-1920.

Wisconsin is a democracy. Up until 1960, the mayor was mostly a socialist. Daniel Hoan, the second socialist mayor, was elected in 1916. Frank Zeider, the third socialist mayor, was elected in 1948.

The socialists focused on cleaning up Milwaukee, both figuratively and literally. For the former, they fought corruption and graft. They obtained their name, Sewer Socialists, from the latter. Introducing sanitation, electric, and power systems was an importance advance in their day.

The name, I guess, was coined by Morris Hillquit, a prominent member of the Socialist Party of America (1901 – 1972). He was tired of the boasting from Milwaukee socialists. But they had good cause to boast.

As I understand it, German immigrants, who fled to the US after participating in the failed 1848 revolutions, were an important current for the development of the sewer socialists. Nicholas Howland, in the link, makes an explicit link to Eduard Bernstein's reformism. Bernie Sanders, the former mayor of Burlington, Vermont, can be said to be in this tradition. Ruth Messinger, a former city council member in New York City, borough president of Manhattan, and co-chair of the Democratic Socialists of America is maybe another example.

Extensive Rent And Labor Values

1.0 Introduction

Do scarce natural resources provide additional difficultes for modern reconstructions of classical and Marxian theories of value? After all land can be sold or rented, and labor cannot produce more land. (I put aside Holland.)

This post presents an exposition of the theory of extensive rent, a start on examining possible difficulties. This type of rent provides the least dificulties, as I understand it, for such modern reconstructions. As usual, I present an example, close to the minimal complexity, needed to make my points. The model can obviously be generalized to include many more produced industrial commodities; many more types of agricultural commodities; and many more types of land, each specialized to support the production of one kind of agricultural commodity.

2.0 Technology

Table 1 specifies the technology for this example. Each column defines the coefficients of production for a process. For example, the only iron-producing process requires a_0,1 person-years of labor, a_1,1 tons of iron, and a_2,1 bushels of corn as inputs for every ton iron produced. I assume that each process requires a year to complete and exhibits constant returns to scale. The corn-producing processes each have an upper limit on how much corn they can produce.

Table 1: A Technology

	Iron Industry	Corn Industry
	Process a	Process b	Process c
Labor	a0,1	a0,2	a0,3
Land, Type 1	c1,1 = 0	c1,2 > 0	c1,3 = 0
Land, Type 2	c2,1 = 0	c2,2 = 0	c2,3 > 0
Iron	a1,1	a1,2	a1,3
Corn	a2,1	a2,2	a2,3
OUTPUTS	1 ton iron	1 bushel corn	1 bushel corn

I assume two types of land exist, distinguished by the processes that can be operated on them. A single corn-producing process can be operated on each type of land. Only a certain number of acres of each type of land exists. Each corn-producing process leaves the land unchanged at the end of operating the process. The given quantities of land limit how much corn can be produced. This model cannot accomodate a positive steady-state rate of growth without technical progress.

A full specification for this model should include requirements for use. I assume that the net output must be such that both types of land are farmed, but only one type is fully farmed. Two techniques for production exist, as shown in Table 2. All three processes are operated in each technique, but only one type of land is fully used.

Table 2: Specification of Techniques

Technique	Type 1 Land	Type 2 Land
Alpha	Partially farmed	Fully farmed
Beta	Fully farmed	Partially farmed

3.0 Parameters and Variables

I have already implicitly defined certain parameters above. Table 3 lists certain parameters I use in this model. Table 4 lists variables that I need. Some assumptions are imposed on the matrices A_α and A_β:

• All produced commodities are basic. Iron and corn enter directly or indirectly into the production of both commodities.
• The technology expressed by these matrices is productive. Each matrix satisfies the Hawkins-Simon condition.

Table 3: Selected Parameters

Symbol	Definition
a0, α	Two-element row vector consisting of first two labor coefficients.
a0, β	Two-element row vector consisting of first and third labor coefficients.
Aα	2x2 matrix, with columns consisting of iron and corn coefficients of production for first and second processes.
Aβ	2x2 matrix, with columns consisting of iron and corn coefficients of production for first and third processes.
d	Two-element column vector consisting of iron and corn quantities in the numeraire.

Table 4: Variables

Symbol	Definition
vα	2-element row vector of labor values when type 1 land is free.
vβ	2-element row vector of labor values when type 2 land is free.
p	2-element row vector of prices of unit quantities of iron and corn.
p1	The price of iron, in numeraire units per ton. The first element of p.
p2	The price of corn, in numeraire units per bushel. The second element of p.
rho1	The rent of type 1 land, in numeraire units per acre.
rho2	The rent of type 2 land, in numeraire units per acre.
w	The wage, in numeraire units per person-year.
r	The rate of profits.

4.0 Labor Values

Given the technique in use, how much additional labor would be employed throughout the economy if the net output was such that one additional unit of iron were produced? This is the labor value of iron, and it easily calculated in the theory. The answer to the same question for corn is its labor value.

Suppose type 1 land is free. Then labor values are:

v_α = a_{0, α} (I - A_α)^-1

Labor values, when type 2 land is free, are the corresponding Leontief employment multipliers for the Beta technique. Variations in net output require varying the amount of the land farmed on the type of land that is not fully farmed.

5.0 Prices of Production

With market prices, some operated processes will be obtaining a higher rate of profits than average, and some will be obtaining a lower rate. These variations in the profit rates are perhaps a signal to capitalists that they should disinvest in some industries or processes and increase investment in others. Models of cross-dual dynamics and other models explore these disequilibria.

Prices of production are such that these signals are absent. All operated processes obtain the same rate of profits. I assume profits, rents, and wages are paid out of the surplus product at the end of the year. The following three equations express the condition that all processes obtain the same rate of profits:

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

(p₁ a_1,2 + p₂ a_2,2)(1 + r) + rho₁ c_1,2 + w a_0,2 = p₂

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

The next equation expresses the condition that the price of the numeraire is unity:

p d = 1

Finally, one of the rents must be zero:

rho₁ rho₂ = 0

The last equation is a defining feature of the theory of extensive rent.

Suppose one of the types of land is rent-free. For deiniteness, let type 1 land be only partially farmed. Then the first four equations are in terms of five variables (p₁, p₂, rho₂, w, r). Just as in the case with only circulating capital, prices of production are specified up to one degree of freedom. In classical political economy, the wage is take as given.

6.0 Choice of Technique

Suppose the wage is non-negative and does not exceed a maximum defined by the technology. The system of equations for prices of production has two solutions. Each solution has the rent on one type of land set to zero. The cost-minimizing technique is the one in which the rent on the other land is positive. If, for a technique, the rent on a type of land is negative, that technique will not be adopted by capitalists. At a switch point, the rents on both types of land are zero.

But the analysis of the choice of technique can be expressed in terms of wage curves. Suppose rents were zero. Consider the first two equations in the system of equations for the prices of production and the equation setting the price of the numeraire to unity. These equations yield a function in which the wage decreases with an increase in the rate of profits. Similarly, the first and third equations yield another decreasing wage curve.

In the case of circulating capital alone, the cost-minimizing technique is found by the wage frontier formed out of the outer envelope of these wage curves. At a given wage, the cost-minimizing technique maximizes the wage.

In this example of extensive rent, the cost-minimizing technique is found by the wage frontier formed out of the inner envelope of the wage curves.

In either case, the appropriate wage frontier shows that a lower rate of profits is associated with a higher wage and vice versa. The maximum wage occurs when the rate of profits is zero. The maximum rate of profits arises when the wage is zero.

7.0 Special Cases

Which land is free and which land pays a rent depends on either the wage or the rate of profits, whichever is taken as exogenous in the system of prices of production. At any rate, a wage frontier exists in which the wage is higher the smaller the rate of profits. This frontier is not the outer frontier of the wage curves for the technique.

Without loss of generality, suppose the Alpha technique is cost-minimizing. Type 1 land is not fully farmed and pays no rent. Then labor values are defined, based on the iron-producing process and the process on type 1 land.

Consider the special case in which a_{0, α} is an eigenvector corresponding to the maximum eigenvector for A_α. Then relative prices of production are equal to relative labor values.

On the other hand, suppose that the numeraire is the standard commodity, as found from a_{0, α} and A_α. Suppose only the standard commodity is produced. In this case, only the process on the rent-free land would be used, in contradiction to the analysis of the choice of technique. And suppose the wage is paid out in the form of the standard commodity. Then the following hold:

• The labor value of gross output is equal to total gross output, evaluated at prices of production.
• The labor value of net output is equal to net output, evaluated at prices of production.
• The labor value of the proportion of the standard commodity paid out in wages is equal to wage goods, evaluated at prices of production.

This special case seems especially forced in the case of extensive rent. Is some reformulation available in which surplus value can be treated as the sum of profits and rent?

I do not address the use of labor values in Marx's account of exploitation, Marx-biased technical change, and so on. The special cases in which the labor theory of value hold make obvious that, for a given technology, a higher rate of profits require a lower wage. And this wage frontier continues to hold in models of extensive rent.

8.0 Conclusion

The inclusion of natural resources, insofar as they can be modeled by extensive rent, does not seem to pose any additional issues for modern formulations of classical and Marxian political economy. It does highlight some issues that arise in models with circulating capital.

Labor values can be calculated for all produced commodities, given the technique in use. They are calculated from the marginal land that receives no rent. But suppose that a choice of technique exists. Then, an analysis at the level of prices of production must be prior to the calculation of labor values. The theory of extensive rent highlights this issue.

As Ricardo and Marx noted, prices of production are generally not proportional to labor values. They are equal in the special case, in which all industries have equal organic compositions of capital, in both models of circulating capital and of extensive rent. In the latter case, the organic composition of capital is found for agriculture from no-rent lands partially farmed.

A commodity of average organic composition is picked out in both models. Total labor values and the labor value of wages are equal to the corresponding aggregates in the system of prices of production when this average commodity is used as numeraire and is produced. These invariants, though, have to restricted to the production of the numeraire with the iron-producing process and the process on no-rent land. It is not clear to me that Marx thought his invariants held in his chapters on rent, given their location towards the end of volume 3 of Capital.

Obviously, these observations on natural resources and rent are just a start. They do seem to match what Ricardo was about in the second chapter of his Principles. The analysis of the choice of technique can be thought of, somewhat, as a critique of Ricardo.

At any rate, prices of production are well-defined in models of extensive rent. And they can be used in an analysis of the choice of technique. As usual, I present the analysis with no mention of utility maximization, preferences, or tastes.

Elsewhere

• Branko Milanovic reviews Nat Dyer's Ricardo's Dream.
• Ben Burgis explains Why Marx is back in fashion.
• Symposium on Michael Heinrich's work.
• Doug Henwood interviews John Cassidy on his new book, Capitalism and Its Critics: A Battle of Ideas in the Modern World. (Apparently there are overviews of critics I have never heard of.)
• James Surowiecki Interviews John Cassidy, also on his new book.
• Nathan Robinson interviews Richard Wolff.

Even More On Recurrence Of Truncation Without Reswitching

Figure 1: Partitions in a Parameter Space with Small a1, 1, Small a1, 3

1.0 Introduction

I have been exploring patitions of parameter spaces by fluke switch points. I always have trouble visualizing higher dimensional spaces. Sometimes, my examples are two-dimensional. If I were doing more formal mathematics, instead of numerical exploration, I would not need to care.

This post is a continuation of the example developed here, here, and here. I present two-dimensional slices of a four-dimensional space.

2.0 Perturbations of Coefficients of Production

As the result of technical change, coefficients of production vary. No variation in labor coefficients or of the output matrix are considered here so as to retain reverse labor substitution at the switch point between Beta and Delta and forward substitution of labor at the switch points between Alpha and Gamma and between Gamma and Delta. Accordingly, consider perturbations of the coefficients of production in the first row of the input matrix. These parameters define a four-dimensional space. How does the analysis of the choice of technique vary with a decrease in these coefficients? The recurrence of truncation turns out be only a transient possibility in secular time with this specific model of technical change. The reverse substitution of labor can occur without the recurrence of truncation, but is also transient. Capital reversing, also transient, occurs in one region of parameter space.

2.1 Small Amounts of Circulating Capital Needed for New Machines

The parameter space is partitioned by parameters corresponding to fluke switch points. Figure 1 shows partitions with the values of a_1,1 and a_1,3 as in this previous post. A switch point is a fluke if it is a knife edge case in which almost all perturbations of model parameters destroy its defining properties. Four of the five partitions shown correspond to a switch point on the axis for the rate of profits. With four techniques, switch points between six pairs are possible. A switch point between Alpha and Delta can only occur as an intersection of all four curves. The same goes for a switch point between Beta and Gamma. These two pairs of techniques correspond to the partition for the switch point at which all four wage curves intersect, called here a four-technique pattern.

The sequence of techniques along the wage frontier is invariant in each numbered region. Table 1 lists the cost-minimizing techniques in each region, in order of an increasing rate of profits. Consider perturbations of the coefficients of production, a_1,2 and a_1,4, that specify the inputs of corn needed for each process in which the machine is run for the second year of its life. In region 1, to the northeast, old machines require so much circulating capital to operate that it is never cost-minimizing to run them for a second year. The Alpha technique is cost-minimizing, whatever the distribution of income. An improvement in the efficiency of old machines in using circulating capital in the machine industry leads to machines being operated for two years in that industry at large rates of profits, as in region 5. A similar improvement in the efficiency of old machines in the corn industry leads to machines being operated for two years in the corn industry, as in region 2. The recurrence of truncation only occurs in region 4.

Table 1: Overview of Regions

Region	Techniques	Notes
1	Alpha	No switch point.
2	Alpha, Gamma	Lower rate of profits associated with truncation in corn industry, greater output per worker.
3	Alpha, Gamma, Delta	Lower rate of profits associated with truncation, greater output per worker.
4	Alpha, Gamma, Delta, Beta	Recurrence of truncation in corn industry.
5	Alpha, Beta	Lower rate of profits associated with truncation in machine industry, greater output per worker.
6	Gamma, Delta	Lower rate of profits associated with truncation in machine industry, greater output per worker.
7	Gamma	No switch point.
8	Beta	No switch point.
9	Delta	No switch point.
10	Delta, Beta	Lower rate of profits associated with extension of economic life in corn industry, reverse substitution of labor.
11	Gamma, Delta, Beta	Lower rate of profits associated with truncation in machine industry, extension of economic life in corn industry, reverse substitution of labor.
12	Alpha, Beta, Delta	Around the Beta vs. Delta switch point, lower rate of profits associated with truncation, smaller output per worker.
13	Alpha, Beta, Delta	Lower rate of profits associated with truncation, greater output per worker.

2.2 Large Amounts of Circulating Capital Needed for New Machines in the Corn Industry

Figure 2 shows the partitions in the parameter space at a higher level of a_1,3. The structure in Figure 1 has moved upwards and to the left. The wedge in which the recurrence of truncation appears, region 4, has widened a bit. A new partition has appeared, for a fluke switch point between Gamma and Delta on the wage axis. For parameters where the partitions between regions 2 and 3 and between regions 2 and 7 intersect, the wage frontier has fluke switch points on both the wage axis and the axis for the rate of profits. For a low enough value of a_1,4, it is no longer cost-minimizing to truncate the machine in both industries at a low rate of profits. Gamma is cost-minimizing, whatever the distribution of income, for a high enough value of a_1,2 and a low enough value of a_1,4.

Figure 2: Parameter Space with Small a1, 1, Large a1, 3

2.3 Large Amounts of Circulating Capital Needed for New Machines in the Machine Industry

Figure 3 shows the partitions in the parameter space at a higher level of a_1,1. The amount of corn needed to operate a new machine in the corn industry, a_1,3, is the same as in Figure 1. Only the partitions and regions to the right are labeled in the figure. This part of the parameter space resembles those slices examined above, with a couple of variations. To the right, the lower boundary of region 5 corresponds to a switch point on the axis for the rate of profits, not a fluke switch point in which all four wage curves intersect. The upper boundary of region 3 now is, to the right, such a four-technique pattern of switch points.

Figure 3: Parameter Space with Large a1, 1, Small a1, 3

Figure 4 is an enlargement of the lower left of this slice of the parameter space. Three new points that are a fluke in multiple ways have appeared. One point corresponds to the intersection of the four wage curves on the wage axis. This point is the intersection of five partitions in the parameter space, just as the point in parameter space corresponding to the intersection of the four wage curves on the axis for the rate of profits. The region in which the recurrence of truncation occurs, region 4, has appeared. The reverse substitution of labor also occurs in regions 10 and 11. Two points in the part of the parameter space depicted correspond to fluke switch points simultaneously lying on both axes.

Figure 4: Parameter Space with Large a1, 1, Small a1, 3 (Enlarged)

Figure 5 enlarges this slice of the parameter space in the middle of Figure 3. The sequence of cost-minimizing techniques, in order of an increasing rate of profits, is Alpha, Beta, and Delta in both region 12 and region 13. The boundary between regions 12 and 13 corresponds to an intersection of the Beta and Delta wage curves on the waxis but not on the frontier. In region 12, the wage curves for Beta and Delta intersect twice in the first quadrant. The second intersection is on the frontier. It is a switch point exhibiting capital-reversing. Around the switch point between Beta and Delta a lower rate of profits is associated with a decreased capital-intensity and decreased net output per worker in the economy as a whole.

Figure 5: Parameter Space with Large a1, 1, Small a1, 3 (Another Enlargement)

3.0 Conclusion

The exploration of partitions in parameter space supports an analysis of technical change in which the circulating capital needed to operate a machine decreases. A trajectory from the upper right to the lower left in the figures and from Figures 2 or 3 to Figure 1 represent this specific form of technical change.

This analysis illustrates how perturbing coefficients of production affects the analysis of the choice of technique. Parameter spaces are partitioned by fluke switch points into regions in which the variation of the cost-minimizing technique with distribution is itself invariant. Some of the structures formed by intersections of partitions are not specific to the example, or even to models of fixed capital. For example, fluke switch points can arise in many models simultaneously on both the wage axis and the axis for the rate of profits. Perturbations of parameters around such a point eliminates or creates the possibility for certain techniques to be cost-minimizing at extreme ranges for the rate of profits.

Likewise, perturbation analysis supports the analysis of specific forms of technical progress. By contrast with fluke switch points, the recurrence of truncation, the reverse substitution of labor, and capital-reversing are not flukes. The reverse substitution of labor occurs in regions in which the recurrence of truncation does not occur. In this sense, the reverse substitution of labor is more general in this example. Capital-reversing arises in a region in which neither of these other 'perverse' phenomena occur. Configurations of parameters in which these phenomena arise, however, define only a few regions of the parameter space. This observation seems capable of generalization to other 'perverse' phenomena in other examples. Other forms of technical progress can be explored.

On Confusion About Gödel's Theorem, Including In Austrian Economics

1.0 Introduction

Gödel's theorems are often invoked in non-mathematical contexts, sometimes in a very imprecise fashion (Franzen 2005, Raatikaine 2007, Jaimungal). You should be skeptical of what many say about Gödel, including of what I say.

In this post, I look at two examples. One is of a confused economist of the Austrian school. The other is of Wittgenstein, who has been ably defended on this point.

2.0 A Claim About the Economic Calculation Problem

Nguyen (2024) says that somehow Gödel's theorem supports the claim that centralized economic planning is, in principle, impossible. Nguyen is sympathetic to the claim that our minds (brains?) transcend the capabilities of all formal systems. I find this claim dubious, but I have not read Penrose. For purposes of argument, Nguyen assumes, through most of this paper, that the central planner has the information von Mises grants them. Nguyen deliberately puts aside Hayek's concerns about non-articulated, distributed, tacit knowledge.

I have demonstrated that von Mises' argument is invalid. I find I am not original. Cockshott (2010) has done the same. Both of us put forth a linear programming formulation that does not require prices of intermediate goods as part of the data. We differ in our specifications of the planner's objective function. Neither of us are echoing Lange and Lerner's formulations of general equilibrium. As such, I do not see any issues are raised for us by the non-computability of utility functions, preference relations, or general equilibria in which ratios of marginal utilities enter the system of equations.

A polynomial-time algorithm exists for solving linear programs. Linear programs are not undecidable. So Gödel's theorem and issues arising from Turing's work do not seem to have any purchase here.

Furthermore, in practice, only a finite number of numbers would be used in drawing up plans. Real numbers would be approximated, if you can call it that, by IEEE Std. 754. This standard defines floating-point numbers, in single and double precision formats. Your computer does not only fail to represent the full range of real numbers. It also only represents a finite number of integers in words, typically of 32 or 64 bits.

I do not mean to suggest that many practical issues do not arise with central planning Nor am I advocating such. I continue to maintain that von Mises' argument is invalid. And I find Nguyen's paper to be one of those mystical, imprecise invocations of Gödel.

3.0 A Notorious Paragraph from Wittgenstein

Here is Wittgenstein from notes not published until after his death:

"8. I imagine someone asking my advice; he says: 'I have constructed a proposition (I will use "P" to designate it) in Russell's symbolism, and by means of certain definitions and transformations it can be so interpreted that it says: "P is not provable in Russell's system". Must I not say tha this proposition on the one hand is true, and on the other hand unprovable? For suppose it were false; then it is true that it is provable. that surely cannot be! And if it is proved, then it is proved that it is not provable. Thus it can only be true, but unprovable.'

Just as we ask, '"Provable" in what system?,' so we must also ask, '"True" in what system?' 'True in Russell's system' means, as was said, proved in Russell's system, and 'false in Russell's system' means the opposite has been proved in Russell's system. - Now what does your 'suppose it is false' mean? In the Russell sense it means, 'suppose the opposite is proved in Russell's system'; if that is your assumption you will now presumably give up the interpretation that it is unprovable. And by 'this interpretation' I understand the translation into this English sentence. — If you assume that the proposition is provable in Russell's system, that means it is true in the Russell sense, and the interpretation 'P is not provable' again has to be given up. If you assume that the proposition is true in the Russell sense, the same thing follows. Further: if the proposition is supposed to be false in some other than the Russell sense, then it does not contradict this for it to be proved in Russell's system. (What is called 'losing' in chess may constitute winning in another game.)" -- Wittgenstein (1978: Part I, Appendix III)

Wittgenstein seems to equate provability in PM with being true in PM. Is not part of Gödel's point to separate these concepts? Furthermore, Wittgenstein seems to be writing only about the heuristic argument given at the start of Gödel's paper. Do his remarks make sense of Gödel numbering, (primitive) recursive functions, and so on?

As I understand it, Gödel's proof of his incompleteness theorem is a conventional proof. The proof is an argument to convince you that a sequence of statements exists that follow one another, by conventional deduction rules. And the conclusion is a theorem about natural numbers.

Gödel's proof is about the syntactic manipulation of strings of symbols by formal rules. Wittgenstein questions interpretations, I guess, of Godel numbers as the statements or sequence of statements that map into them. Floyd and Putnam (2000) justify Wittgenstein in a way that draws on the distinction between consistency and ω-consistency.

J. B. Rosser replaced ω-consistency with consistency in Gödel's proof. A theory is ω-inconsistent if one can show, for some proposition p:

• not p(0), not p(1), not p(2), ..., and
• There exists x such that p(x).

I guess that a non-standard model can be ω-inconsistent, where x is not a natural number, somehow. Suppose PM is ω-inconsistent. And suppose the negation of the Gödel sentence 'P' is true. Then the Godel number for the proof of this proposition could be a non-standard natural number. Wittgenstein writes about interpretations, but does not mention ω-consistency. I suppose he could have known about the Löwenheim-Skolem theorem.

I am sympathetic to being suspicious of English-language interpretations of syntactical manipulations. I am not so sympathetic to how lightly Wittgenstein treats possible contradictions in mathematics. I am sympathetic to the idea that mathematical logic, set theory, and model theory, for example, just provides more maths. One does not need this math to justify what humans have been doing for millennia. Do mathematical propositions have meaning before proofs are found? As I understand Wittgenstein, he says not. Proofs draw connections and give the proved proposition a meaning.

But I am also aware that even if I can echo out some propositions from these fields, I am no expert in them.

References

• Cockshott, W. Paul. 2010. Von Mises, Kantorovich and in-natura calculation. European Journal of Economics and Economic Policies 7(1): 167-199.
• Dawson, Ryan. 2015. Leaving Mathematics As It Is: Wittgenstein's Later Philosophy of Mathematics. University of East Anglica. PhD Thesis.
• Floyd, Juliet and Hilary Putnam. 2000. A note on Wittgenstein’s 'notorious paragraph' about the Gödel theorem. Journal of Philosophy. 97(11): 624-632.
• Franzen, Torkel. 2005. Gödel's Theorem: An Incomplete Guide to Its Use and Abuse. A. K. Peters.
• Jaimungal, Curt. Theory of Everything: a podcast episode.
• Nguyen, Hai-Trieu. 2024. The incompleteness of central planning. Quarterly Journal of Austrian Economics. 27(4).
• Raatikaine, Panu. 2007. Review of Franzen. Notices of the AMS. 54(3): 380-383.
• Standford Encylopedia of Philosophy, Wittgenstein's philosophy of mathematics.
• Wittgenstein, Ludwig. 1978. Remarks on the Foundation of Mathematics, revised edition. MIT Press.

Technical Change and Triple-Switching in the Corn-Tractor Model

I have another working paper, Technical change and triple-switching in the corn-tractor model, at the Centro Sraffa. The abstract follows.

Abstract: With triple-switching, each of two techniques are cost-minimizing in two disjoint intervals of the wage or rate of profits. Technology that supports multiple switch points between two techniques can only be a temporary phenomenon, as one technique supplants another with technical progress. A perturbation analysis of a triple-switching example in the corn-tractor model illustrates this claim. A parameter space, defined by two selected coefficients of production, is partitioned by loci corresponding to fluke switch points. The analysis of the choice of technique does not qualitatively vary within each of the resulting regions. Technical progress corresponds to specific trajectories through this parameter space.

Keywords: Cambridge Capital Controversy; Fixed Capital; Reswitching of techniques

JEL Codes: B51: Sraffian; C67: Input-Output Models; D24: Production, Cost, Capital

Howard Fast Leaves The Communist Party Of The USA After Khrushchev's Secret Speech

This post is another instance of me stumbling over anti-communist literature, in a sense.

I read Howard Fast's novel, Citizen Tom Paine, many years ago. I have never read Spartacus or seen the movie. I think I read April Morning in high school. I think I read Freedom Road at some point. At any rate, I have the impression that reconstruction was actually a moment when it seemed America was coming closer to living up to its promise to provide liberty for all. I also read The Last Frontier at some point.

I believe I knew he was blacklisted. I hadn't known that he actually was a member of the CPUSA. Nor had I known about Peekskill or his time in prison.

Nikita Khrushchev gave his secret speech in February 1956. In it, he denounces the cult of personality around Stalin. I believe this is where the phrase comes from. It looked like that the Soviet Union might fulfill its promise. But by the end of that year, the Soviet Union invaded Hungary to put down their revolution.

The secret speech caused Fast all sorts of struggle. When he left the communist party, he wrote a book explaining his reasons.

"I had a god who walked naked, but nobody among those I loved said so; for even as the innocent wisdom of Hans Christian Andersen held that those who could not see the king's clothes were persons of small intellect and unfit for the positions they held, so in my world, it was the conviction of millions of good and wise folk that only those who had lost all honor, dignity, decency and courage would dare to point out that this god whom we worshiped for his noble raiment was indeed naked and ugly in his nakedness.

Who would be without honor, dignity, decency and courage?

In the little town where I live, there is a little store, unimportant and of no consequence, and out of this store an old man ekes a living. This is an old man who mourns a hurt which will not heal, the kind of hurt many who read this will know intimately, for twenty years ago the young son of this man fell in Spain, fighting in the Lincoln Battalion for the Spanish Republic and the freedom of men. His son lies buried in the distant Spanish soil, and for twenty years the hurt in this old man was as if it had happened yesterday.

He had a little salve to rub on the terrible sore. This was the salve, that his son had died in the best of causes, the fight for the liberation of mankind. But in 1956, a man called Khrushchev delivered a certain 'secret report' - telling a story of Russia and the Communist movement that I and my friends had heard before but had never believed before. Now Khrushchev made proof of twenty'five years of 'slander,' and we believed. And among those who believed because they had to believe was this old man whose son had laid down his life in Spain.

I came into his store one day in that month of June and he was weeping. He asked me,

'Why did my son die?'

For had I not held, all of my thinking life and in all that I wrote, that one son of man was all the sons of man? He then said to me, but not in words - for a broken heart does not make a gentle person cruel or vindictive - not in words but with the look in his eyes -

'That I, a plain man did not comprehend this is no wonder; but you, Howard Fast, spoke and wrote and pleaded this cause - and why? Can you tell me why?'" -- Howard Fast, The Naked God: The Writer and the Communist Party, 1957.

None of this means Fast did not keep the same values:

"I am neither disillusioned nor depressed, and I have lived through grim times, but times when mankind made gigantic strides forward. Though the Communist Party is disciplined and often splendid in military action, I do not think it can claim credit for the events we have seen. Socialism, justice and the brotherhood of man are mighty and irresistible forces; they will grow to fruition in spite of the Communist Party - and Soviet socialism will not forever lie supine under the heel of the commissar."-- Howard Fast, chapter 15.

Capital-Reversing In A Pertubation Of An Example Of The Recurrence Of Truncation

Figure 1: The Wage Frontier for an Example of Capital-Reversing

I am continuing to explore perturbations of coefficients of production for inputs of circulating capital in this example. The example is of the recurrence of truncation without either the reswitching of techniques or capital reversing. This post presents a perturbation in which the recurrence of truncation no longer occurs, but capital-reversing now arises.

Tables 1 and 2 present the coefficients of production for the example. The technology has the minimum structure, in a model in which multiple commodities are produced, and inputs of labor, circulating capital, and fixed capital are needed in each industry. Also, managers of firms in both industries have a choice of the economic life of a machine. In this perturbation, a_{1, 1} is higher than in the example I started with. a_{1, 3} is the same. I varied a_{1, 2} and a_{1, 4} to find this example of capital-reversing.

Table 1: Inputs for The Technology

Input	Industry
	Machine		Corn
	I	II	III	IV
Labor	1/10	8	43/40	1
Corn	32/50	2/5	1/8	0.38944767
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

Table 3 repeats the definition of the available techniques. I will observe that a switch point between Beta and Gamma cannot exist without that switch point being simultaneously a point at which all wage curves intersect.

Table 3: Specification of Techniques

Technique	Processes
Alpha	I, III
Beta	I, II, III
Gamma	I, III, IV
Delta	I, II, III, IV

Figure 1, at the top of this post, depicts the wage curves for the four techniques. It is not visually striking. In particular, the wage curves for Beta and Delta are hard to distinguish by eye. Figure 2, below, is an enlargement of part of the wage curves so that you can see that Beta and Delta have different wage curves. The sequence of wage curves on the frontier, in order of an increasing rate of profits, is Alpha, Beta, Delta. Gamma is never on the frontier.

Figure 2: The Wage Frontier for an Example of Capital-Reversing (Enlargement)

This example is one of my usual illustrations that the Austrian and marginalist theories of capital are muddled and confused. Around the switch point between Beta and Delta, a lower interest rate is associated with the truncation of the machine in the corn industry and a less-capital intensive technique. Net output per worker has decreased for the economy as a whole. Around the switch point between Alpha and Beta, a lower interest rate is associated with the truncation of the machine in the machine industry and a more capital-intensive technique. Net output per worker is increased for the economy as a whole.

A lower interest rate need not encourage capitalists to adopt more capital-intensive techniques. No necessary association exists with extending the economic life of a machine and adopting a more capital-intensive technique.