Instability, A Problem For The Theory Of Supply And Demand

General Equilibrium Theory (GET) is the most rigorous and most developed version of the theory of supply and demand. More than half a century ago, economists discovered that GET does not support the just so stories many economists still tell. (Experimental economists have validated Scarf's example of instability.)

Franklin Fisher was an authority on this topic.

"Yet the very power and elegance of equilibrium analysis often obscures that it rests on a very uncertain foundation. We have no similarly elegant theory of what happens out of equilibrium, of how agents behave when their plans are frustrated. As a result, we have no rigorous basis for believing that equilibria can be achieved or maintained if disturbed. Unless one robs words of their meaning and defines every state of the world as an 'equilibrium' in the sense that agents do what they do instead of doing something else, there is no disguising the fact that this is a major lacuna in economic analysis.

Nor is that lacuna only important in microeconomics. For example, the Keynesian question of whether an economy can become trapped in a situation of underemployment is not merely a question of whether underemployment equilibria exist. It is also a question of whether such equilibria are stable. As such, its answer depends on the properties of the general (dis)equilibrium system which macroeconomic analysis attempts to summarize. Not surprisingly, modern attempts to deal with such systems have been increasingly forced to treat such familiar macroeconomic issues as the role of money." -- Franklin M. Fisher. 1987. Adjustment processes and instability. In J. Eatwell, M. Milgate, P. Newman, The New Palgrave: A Dictionary of Economics. Macmillan.

As far as I know, although some interesting work has been done, this gap still exists.

Reswitching In A Model With Exhaustible Resources?

1.0 Introduction

Part of my program is to construct numeric examples of the reswitching of techniques and of other capital-theoretic 'paradoxes' in a variety of models. Models of exhaustible resources, under some interpretation, provide another opportunity.

2.0 Background

A dispute exists on how compatible the existence of exhaustible resources is with the long period method. An excessively simple analysis is to treat payments for exhaustible resources like rent paid for 'the original and indestructible powers of the soil' (Ricardo 1951: 67). At a given point in time, the cost of mining a resource varies among deposits, and rent varies among mines in use. More than one mine is typically operated for an exhaustible resource because of constraints on the capacity at which mine can be operated. This, arguably, is Ricardo's approach in Chapter III, on the rent of mines, in his Principles. Sraffa (1960), too, groups 'land and mineral resources' together, under the heading of natural resources. Kurz & Salvadori (2009, 2015) read both Ricardo and Sraffa as having a somewhat more sophisticated approach

Parrinello (1983) and Schefold (1989) consider exhaustible resources as such. Bidard & Erreygers (2001, 2020), with the corn-guano model, argue that exhaustible resources are inconsistent with the long period method. The story associated with the model is about an island which was populated by dodos. Manure increases the yield of corn. But, since the dodo is extinct, no more guano is being made in the story. Eventually, capitalist farmers must adopt the backstop technology. They argue that a royalty for an exhaustible natural resource will vary over time, in accordance with the Hotelling rule. All prices will vary over time, as long as exhaustible resources are used in production

Parrinello (2004) and Kurz & Salvadori (2009, 2011, 2015) argue that when the resource will be exhausted is not well-enough known for the Hotelling rule to fully apply. In Kurz & Salvadori's models, a constraint on mines limits how much of each exhaustible resources can be brought above ground in each production period. The price of unmined exhaustible resources increases in accordance with the Hotelling rule, but the rent on mines can decrease in parallel, leaving the price of mined resources unchanged over time. Even though this is an intertemporal model, the prices of produced commodities do not vary over time.

For Ravagnani (2008), the royalty for an exhaustible resource provides another degree of freedom and is set as a percentage of production by conventions and social norms, much like the natural wage in Ricardo and Marx. Huang (2018) builds on Kurz & Salvadori and treats exhaustible resources by introducing processes to search for resources.

3.0 Parameters

Tables 1 and 2 specify a technology that extends an example from Kurz & Salvadori (2011) to include the production of iron. Iron and corn are basic commodities in the sense of Sraffa. Each column in Table 1 specifies the inputs needed to operate the process at a unit level. Each column in Table 2 specifies the corresponding outputs for the process, when operated at a unit level. All processes exhibit constant returns to scale (CRS).

Table 1: Inputs for The Technology

Output	Process
	I	II	III	IV	V	VI	VII	VIII
Labor (Person-Yrs.)	a0,1	a0,2	a0,3	a0,4	a0,5	a0,6	-	-
Iron (Tons)	a1,1	a1,2	a1,3	a1,4	-	-	-	-
Corn (Bushels)	a2,1	a2,2	a2,3	a2,4	-	-	-	-
Oil Underground (Barrels)	-	-	-	-	a3,5=1	-	a3,7=1	-
Menthane Underground (K-Litres)	-	-	-	-	-	a4,6=1	-	a4,8=1
Extracted Oil (Barrels)	-	-	a5,3=1	-	-	-	-	-
Extracted Menthane (K-Litres)	-	-	-	a6,4=1	-	-	-	-

Table 2: Outputs for The Technology

Output	Process
	I	II	III	IV	V	VI	VII	VIII
Iron (Tons)	b1,1 = 1	-	-	-	-	-	-	-
Corn (Bushels)	-	b2,2=1	b2,3=1	b2,4=1	-	-	-	-
Oil Underground (Barrels)	-	-	-	-	-	-	b3,7=1	-
Menthane Underground (K-Litres)	-	-	-	-	-	-	-	b4,8=1
Extracted Oil (Barrels)	-	-	-	-	b5,5=1	-	-	-
Extracted Menthane (K-Litres)	-	-	-	-	-	b6,6=1	-	-

The first process produces iron. The next three processes produce corn. The first corn-producing process is part of a backstop technology. The other two use oil and menthane as fertilizer, respectively. Processes V and VI are extraction processes. Processes VII and VIII are conservation processes.

The extraction processes, V and VI, are constrained not to produced more than a maximum output in any year. I let t_o be the maximum output for oil extraction in a year. Let t_m be the maximum output for menthane extraction in a year. The data also includes the specification of q_uo, the initial quantity of unexextracted oil, and q_um, the initial quantity of unextracted menthane.

Final demand, y₁ and y₂ for iron and corn, are the last parameters needed to specify this model.

4.0 Selected Price Variables

I need to specify techniques and quantity flows for each technique. The amount of unextracted oil and methane will vary over time. Depending on the parameters, only some techniques will be feasible.

Table 3 defines the price variables that will be found by solving the price equations. Rent on mines and the prices of unextracted exhaustible resources vary over time, as reflected in the notation.

Table 3: Selected Price Variables

Variable	Definition
p1	Price of a ton iron.
p2	Price of a bushel corn.
po	Price of a barrel of extracted oil.
pm	Price of a kilo-litre of extracted methane.
puo(t)	Price of a barrel of unextracted oil at the end of the tth year.
pum(t)	Price of a kilo-litre of unextracted menthane at the end of the tth year.
rhoo(t)	Rent per barrel oil extracted in a year.
rhom(t)	Rent per kilo-litre methane extracted in a year.
w	The wage.
r	The rate of profits.

6.0 Conclusion

My problem is to find numeric values for model parameters such that reswitching results. This reswitching might be analogous to the reswitching of the order of efficiency. Perhaps for some ranges of the rate of profits, the backstop technology is operated along with the extraction of oil. And, at other ranges, the backstop technology is operated along with the extraction of methane.

Or maybe the solution will be that corn is produced with process III, without the backstop technology, at some ranges of the rate of profits. And, at other ranges, corn is produced with process IV, also without the backstop technology.

This post only poses a problem. I do not think it do difficult to see that capital-theoretic 'paradoxes' can appear in Kurz and Salvadori's approach to exhaustible resources. I suppose it would be good to have concrete examples.

References

• Bidard, C. and G. Erreygers. 2001. The corn-guano model. Metroeconomica 52(3): 243-253.
• Bidard, C. and G. Erreygers. 2020. Exhaustible resources and classical theory. History, Methodology, Philosophy 10(3): 419-446.
• Huang, B. 2018. An exhaustible resources model in a dynamic input-output framework: A possible reconciliation between Ricardo and Hotelling. Journal of Economic Structures, 7(1): 1-24.
• Kurz, H. D. and N. Salvadori. 2009. Ricardo on exhaustible resources and the Hotelling rule. In Aiko Ikeo and Heinz D. Kurz (eds), A History of Economic Theory: Essays in Honour of Takashi Negishi. London: Routledge.
• Kurz, H. D. and N. Salvadori. 2011. Exhaustible resources: Rents, profits, royalties and prices. In Volker Caspari (ed.), The Evolution of Economic Theory: Essays in Honour of Bertram Schefold. London: Routledge, 39-52.
• Kurz, H. D. and N. Salvadori. 2015. The 'classical' approach to exhaustible resources. Parrinello and the others. In Heinz D. Kurz and Neri Salvadori (eds), Revisiting Classical Economics. Studies in Long-Period Analysis. London: Routledge, 304-316.
• Parrinello, Sergio. 1983. Exhaustible natural resources and the classical method of long-period equilibrium, in J. Kregel (ed.), Distribution, Effective Demand and International Economic Relations, London: Macmillan, pp. 186–99
• Parrinello, Sergio. 2004. The notion of effectual supply and the theory of normal prices with exhaustible resources. Economic Systems Research, 16(3): 311-322.
• Ravagnani, Fabio. 2008. Classical theory and exhaustible natural resources: notes on the current debate. Review of Political Economy, 20(1).
• Ricardo, David. 1951. The Works and Correspondence of David Ricardo: Volume 1: On the Principles of Political Economy and Taxation. Cambridge: Cambridge University Press.
• Schefold, Bertram. 1989. Mr. Sraffa on Joint Production and Other Essays, Routledge.
• Sraffa, Piero. 1960. The Production of Commodities by Means of Commodities: A Prelude to a Critique of Economic Theory. Cambridge: Cambridge University Press.

The Production Of Commodities And The Structure Of Production

I have a working paper at the Munich Personal RePEc Archive (MPRA). MPRA does not have a peer-review process. Here is the abstract:

Abstract: Consider a model of the production of commodities by means of commodities. This article illustrates how to construct a Hayekian triangle with such a model, in the case with circulating capital. Capital-theory paradoxes, specifically the reswitching of techniques, have implications for such triangles. The use of such triangles to tell the stories that Austrian economists want to tell cannot be sustained. The switch point that is normal in a reswitching example, from a mainstream neoclassical perspective, has perverse Hayekian triangles. The switch point that is perverse from a mainstream perspective has Hayekian triangles consistent with the Austrian story about how a decreased time preference rotates the triangle to lengthen the structure of production.

The History Of The Soviet Union Is Inconsistent With Marxism

1.0 Introduction

You can tell a history of the Union of Soviet Socialists Republics in which events are inconsistent with Marx's theory of history.

2.0 Bolshevik Revolution

Marx, like Adam Smith and Walt Rostow, had a stages theory of history. Feudalism was succeeded by capitalism, and capitalism is to be succeeded by socialism. Socialism is to arise first in the most advanced capitalist countries. (The theory of history is not my favorite part of Marxist theory.) Russia, in 1917, was a semi-feudal country with peasants as the largest class. I guess China was the same, before Mao. A Marxist would not expect socialism to start successful in either country.

I have some caveats. Marx's 1881 letter to Vera Zasulich speculates on the possibility of socialism in Russia. I am aware that Lenin had an argument about how socialism can start with the weakest link in the age of imperialism. But he still expected his revolution to be supported by revolutions in advanced capitalist countries. And, for a moment he seemed to be right, what with the Spartacist revolution in Germany, the Bavarian Socialist Republic, and Hungary.

3.0 Socialism In One Country

Stalin came up with the idea of socialism in one country. That country was still quite backward, not an advanced capitalist country. Is the championing of the Soviet Union by communists in advanced capitlist countries what Marxists would wnat? The Soviet Union presented an alternative. But shouldn't their priority be building socialism at home? And their cause was weakened when the Soviet Union behaved like other great powers in Hungary in 1956 and in Czechoslovakia in 1968.

4.0 Khrushchev's Secret Speech

Khrushchev denounced Stalin in his 'secret' speech of 25 February 1956. He came up with the concept of the cult of personality. What is that from a Marxist perspective? History is supposed to be determined, ultimately, by contradictions in material conditions. How can the mistakes of one leader be so important?

5.0 Gorbachev's Voluntarism

I do not understand the collapse of the Soviet Union. I think of the fall of the Berlin Wall in 1989 and the dissolution of the Soviet Union in 1991.

As I understand it, Gorbachev and other members of the Politburo acquiesced in the dismantling of the Soviet Union. Poor performance pointed out the need for reforms. Nevertheless, I find it strange that political leaders can decide that the system they preside over has no legitimacy like that.

Steve Paxton provides an account of how the fall of the Soviet Union was a triumph for Marx's theory of history.

Marxist Mathematicians

I have been looking at Marx's notes to himself, late in life, on the calculus. Marx relied on an out-of-date textbook, J. L. Boucherlat's 1828 Elementary Treatise on the Differential and Integral Calculus, as well as other out-of-date primary texts.

The foundations of calculus were a mess at that time. Marx was totally correct about that. Echoing Bishop Berkeley, Marx finds that the mathematicians were operating with fractions of the form 0/0. He tried to make sense out of this.

Unbeknownst to Marx, the foundations were being relaid in his day. I will mention the epsilon-delta definition of a limit, the construction of the real numbers as Dedekind cuts, and Cantor's set theory. I suppose Fourier series goes into this story. Marx never knew about any of this.

Some, sympathetic to Marx, argue that he treated the derivative as an operator.

Some mathematicians have been Marxist and socialists. These views have influenced their activities in developing mathematics, at least to the extent of the settings in which they did their mathematics. For this post, I am not going to sort through mathematicians in the Soviet Union or in China. I limit myself to a few in the United States.

• I know little of Chandler Davis' mathematical work. He lived from 1926 to 2022. The University of Michigan fired him in 1954 for refusing to cooperate with the oppression being practiced by the House UnAmerican Committee (HUAC). He went to jail for six months and then into exile into Canada.
• F. William Lawver was an expert in category theory, including its use to describe Hegelian dialectics. He was dismissed in 1971 partly for his political activities.
• David Schweickart has written a number of books outlining how socialism might be implemented. He is both a philosopher and a mathematician. I am not sure that he is a Marxist.
• I know of Stephen Smale principally through his horseshoe map, which is a canonical model for dynamical systems. He won the Fields Medal and denounced the United States for invading Vietnam.
• Dirk Jan Struik (1894-2000) could not get a job in Holland, partly due to his political commitments. He ended up at MIT. Struik co-founded and taught at the Samuels Adams School, one of several institutions set up to teach workers. These, of course, were illegally shut down by the government. HUAC went after Struik himself, and MIT chose the route of cowardice. He applied Marxist ideas to the sociology of mathematics, a field he helped create. He co-founded Science and Society. He also praised Marx's work on the foundations of calculus.

I will not be surprised if others know of more examples.

Reswitching And Other Capital-Theoretic Paradoxes In A Variety Of Models

Apparently, it is part of my project to construct explicit numeric examples of the rewitching of techniques and capital-theoretic paradoxes in a range of models. Here are some examples I have produced:

• The reswitching of techniques in a capitalist economy with non-competitive markets.
• The reswitching of the orders of efficiency and of rentability in a model with extensive rent.
• The reswitching of techniques in a model with extensive rent.
• The complete reversal of the orders of efficiency and rentability in a model with extensive and intensive rent (See region 11 in the example).
• The reswitching of techniques in a model with extensive and intensive rent and with multiple (two) agricultural commodities.
• The recurrence of truncation and reverse labor substitution without the reswitching of techniques.
• Capital-reversing in an example with fixed capital and scarce land.

This post is one of my summary posts.

I Did Not Invent The Concept Of Fluke Switch Points

I find myself writing about generic switch points, fluke switch points, anomalous switch points, fake switch points, normal switch points, and 'perverse' switch points. I do not seem to have definitions with the precision of those in mathematical analysis.

A switch point is a fluke when any perturbation of some parameters, such as coefficients of production, destroys defining features of the switch point. The concept of a fluke goes back to the 1966 symposium on capital theory in the Quarterly Journal of Economics:

"If, by a fluke more than one switch of technique happened to take place at exactly the same point, the nonzero columns [of the matrix formed by the difference of two Leontief matrices] would be more than one" (Pasinetti 1966: 511).

Other participants recognize this fluke case in which four wage curves intersect at a switch point, with processes replacing one another in two industries:

"'Adjacent" techniques on two sides of a switching point of a switching point will usually differ from each other only with respect to one activity" (Bruno, Burmeister, Sheshinski 1966: 542).

Two wages curves tangent at a switch point is another fluke case. A perturbation leads to either the reswitching of techniques or of one cost-minimizing technique around the rate of profits at which the switch point formerly existed. Other fluke cases arise when a switch point exists at the maximum wage or the maximum rate of profits:

"Cases with multiple roots or cases in which the curves cross only at end points... These ... are cases which one technique can be ignored since it is dominated" (Bruno, Burmeister, Sheshinski 1966: 534).

Pierangelo Garegnani recognizes the possibility of the fluke case with two wage curves tangent:

"The possibility that, at r^* and r^**, the two wage curves touch without intersecting is excluded...” (Garegnani 1966: 567).

Fluke switch points exist in both models of single and joint production. Vienneau (2021) examines fluke switch points in pure fixed capital models, while Vienneau (2022) partitions a parameter space, with fluke switch points, in a model of extensive rent. Vienneau (2024) looks at fluke switch points in a model of non-competitive markets with single production. The characterization of fluke switch points is useful for analyzing structural economic dynamics with a choice of technique.

References

Bruno, Michael, Edwin Burmeister, and Eytan Sheshinski. 1966. The nature and implications of the reswitching of techniques. Quarterly Journal of Economics 80(4): 526-553.

Garegnani, P. 1966. Switching of techniques. Quarterly Journal of Economics 80(4): 554-567

Pasinetti, Luigi L. 1966. Changes in the rate of profit and switches of technique. Quarterly Journal of Economics 80(4): 503-517.

Vienneau, Robert L. 2021. Fluke switch points in pure fixed capital systems. Centro Sraffa working papers n. 48.

Vienneau, Robert L. 2022. Reswitching in a model of extensive rent. Bulletin of Political Economy 16(2): 133-146.

Vienneau, Robert L. 2024. Characteristics of labor markets varying with perturbations of relative markups. Review of Political Economy 36(2): 827-843.