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 Marx relied on an out-of-date textbook, J. L. Boucherlat's 1828 Elementary Treatise on the Differential and Integral Calculus, as well as other out-of-date primary texts.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.

Surveys Of The Cambridge Capital Controversy

'Neoclassical' economists accepted, in the third quarter of last century, that the theories they teach and apply have no rigorous foundation. They are illogical and incoherent. Why does that not matter? This presents a puzzle.

Many have surveyed or responded to the Cambridge Capital Controversy. Here are some surveys and responses:

• Jack Birner. 2002. Cambridge Controversies in Capital Theory: A Methodological Analysis. Routledge.
• Mark Blaug. 1974. The Cambridge Revolution: Success or Failure?. London: Institute of Economic Affairs. (I stumbled upon this negative review in the History of Political Economy.)
• Christopher Bliss. 1975. Capital Theory and the Distribution of Income. Elsevier North-Holland.
• Edwin Burmeister. 1982. Capital Theory and Dynamics. Cambridge University Press.
• Avi J. Cohen and G. C. Harcourt. 2003. Whatever happened to the Cambridge capital theory controversies. Journal of Economic Perspectives 17(1): 199-214.
• Avinash Dixit. 1977. The accumulation of capital theory. Oxford Economic Papers 29(1): 1-29.
• Roger W. Garrison. 2006. Reflections on reswitching and roundaboutness. In Money and Markets: Essays in Honor of Leland B. Yeager (ed. by Roger Koppl). Routledge.
• G. C. Harcourt. 1969. Some Cambridge controversies in the theory of capital. Journal of Economic Literature 7(2): 369-405.
• G. C. Harcourt. 1972, 2022. Some Cambridge Controversies in the Theory of Capital. Cambridge University Press.
• Daniel Hausman. 1981. Capital, Prices and Profits. Columbia University Press.
• Andrés Lazzarini. 2011. Revisiting the Cambridge Capital Theory Controversies: A Historical and Analytical Study. Pavia University Press..
• Joseph E. Stiglitz. 1974. The Cambridge-Cambridge controversy in the theory of capital: a view from New Haven. Journal of Political Economy 82(4): 893-903.
• Leland B. Yeager. 1979. Capital paradoxes and the concept of waiting. In Time, Uncertainty, and Disequilibrium: Exploration of Austrian Themes (ed. by M. J. Rizzo). Lexington Books.

Harcourt (1972) is my favorite of these surveys - an utterly conventional view. I disagree with much in many of these surveys and responses. But those who have never been exposed to the CCC will learn something from any of them.

To Read Adorno's Minima Moralia Requires Understanding Of Marx

Well-known contributions to philosophy in Europe and America in the twentieth century are often divided into analytical and continental philosophy. Analytical philosophers often state their arguments with formal reasoning and notation, while concentrating on narrow points. How do you know that you have not always used 'green' to mean grue? Continental philosophers provide a more intuitive reasoning and focus on larger issues such as culture. Gender is performative. I take no issue to those who argue that the division is not well-defined. I lean more towards the analytical side, albeit I try to reject logical positivism.

Sometimes, when I read postmodernists - another ill-defined term - I can follow, but I do not retain much. I find intriguing Lukacs' essay on reification, in which he builds on Karl Marx's work on commodity fetishism and vulgar political economy. I have more use for Foucault's post structuralism than Derrida's deconstruction. I find Antonio Gramsci insightful, which is no surprise for somebody building on Piero Sraffa. Herbert Marcuse has a point about instrumental reason in the service of a system that is irrational as a whole. I am never sure when Slavoj Zizek is joking.

But here I want to focus on Theodor Adorno and his 1951 book Minima Moralia: Reflections from Damaged Life. I could not make much out of his book, Negative Dialetics. I have yet to read The Dialetic of Enlightenment.

As I poorly recall, Minima Moralia has a narrative arc, although it takes some work to perceive it. I was surprised at passages that presume an understanding of technical terms in Marx's political economy. I note a few here.

Here Adorno rejects some concepts of a post-capitalist society because they continue commodity fetishism:

"Sur l'Eau. He who asks what is the goal of an emancipated society is given answers such as the fulfillment of human possibilities or the richness of life. Just as the inevitable question is illegitimate, so the repellent assurance of the answer is inevitable, calling to mind the social-democratic ideal of the personality expounded by heavily-bearded Naturalists of the 'nineties, who were out to have a good time. There is tenderness only in the coarsest demand: that no-one shall go hungry any more. Every other seeks to apply to a condition that ought to be determined by human needs, a mode of human conduct adapted to production as an end in itself. Into the wishful image of an uninhibited, vital, creative man has seeped the very fetishism of commodities which in bourgeois society brings with it inhibition, impotence, the sterility of the never-changing. The concept of dynamism, which is the necessary complement of bourgeois 'a-historicity', is raised to an absolute, whereas it ought, as an anthropological reflex of the laws of production, to be itself critically confronted, in an emancipated society, with need..." -- Adorno: 155-156.

Adorno draws on the concept of fetishism in other places. I do not know that the above passage is consistent with Marx and Engels in The German Ideology.

In this next passage he uses the concept of the organic composition of capital to write about how working class consciousness is dimmed:

"Puzzle-picture. Why, despite a historical development that has reached the point of oligarchy, the workers are less and less aware that they are such, can be surmised from a number of observations. While objectively the relation of owners and producers to the productive apparatus grows ever more rigid, subjective class membership becomes all the more fluctuating. This tendency is fostered by economic development itself. The organic composition of capital demands, as has often been noted, control through technical experts rather than through factory owners. The latter were the counterpart, as it were, of living labour, the former correspond to the share of machinery in capital. The quantification of technical processes, however, their dissection into minute operations largely independent of education and experience, makes the expertise of these new-style managers to a large degree illusory, a pretence concealing the privilege of being appointed. That technical development has reached a state which makes every function really open to all - this immanently socialist element in progress has been travestied under late industrialism. Membership of the elite seems attainable to everyone. One only waits to be co-opted... ...Preference goes to those who fit in most exactly...That technical forces might permit a condition free of privileges is accredited by all, even those in the shadow, to the social relations which prevent it. In general, subjective class-membership today shows a mobility that allows the rigidity of the economic order itself to be forgotten..." -- Adorno: 193-194.

And, for the last passage I select, Adorno writes about the law of value and, again, the organic composition of capital:

"Novissimum organum. It has long been demonstrated that wage-labour formed the masses of the modern epoch, indeed created the worker himself. As a general principle the individual is not merely the biological basis, but the reflection of the social process; his conciousness of himself as something in-itself is the iI1usion needed to raise his level of performance, whereas in fact the individuated function in the modern economy as mere agents of the law of value. The inner constitution of the individual, not merely his social role, could be deduced from this. Decisive here, in the present phase, is the category of the organic composition of capital. By this the theory of accumulation meant the 'growth in the mass of the means of production, as compared with the mass of the labour-power that vivifies them'. If the integration of society, particularly in totalitarian states, designates subjects more and more exclusively as partial moments in the network of material production, then the 'alteration of the technical composition of capital' is prolonged within those encompassed, and indeed constituted, by the technological demands of the production process. The organic composition of man is growing. That which determines subjects as means of production and not as living purposes, increases with the proportion of machines to variable capital... Only when the process that begins with the metamorphosis of labour-power into a commodity has permeated men through and through and objectified each of their Impulses as formally commensurable variations of the exchange relationship, is it possible for life to reproduce itself under the prevailing relations of production..." -- Adorno: 228-229.

My understanding of the organic composition of capital is straightforward. I take it to be the ratio of constant capital to variable capital, evaluated either with labor values or with prices of production. I do think about the physical composition of capital goods and of issues associated with depreciation. But I certainly do not go into the cultural effects that Adorno writes about.

Numerical Examples Of The Reswitching Of Techniques In Spatial Economics?

It is my custom to work through, with graphs, example of the reswitching of techniques and other capital-theoretic 'paradoxes' in various models. Sometimes I have created numeric examples, and perturbed them too.

Reswitching examples exist in the literature on spatial, regional, and urban economics. Nowhere have I worked through them.

I think of regional economics as having two traditions for building theoretical models. One is the the Von Thunen model, with a central city and concentric rings of land uses. Transport costs are of importance. Reswitching is manifested by non-adjacent rings being used to produce the same commodities, with other commodities being produced in between. Barnes & Sheppard (1984) have graphs suggesting that they have a concrete numerical example. But they do not present parameter values. Their text suggests that an example can be created based on the example in Metcalfe & Steedman (1979).

Walter Isard invented regional economics in the middle of the twentieth century. This approach has different countries or regions described by individual Leontief matrices. Imports and exports show interactions between the regions.

Pavlik (1990) has a numeric example for this second tradition, reproduced, I think, in section 5.1 of Sheppard & Barnes (1990). They have the production distributed among regions, as I understand it, like in 'gravity models'. And its solution requires an application of an iterative algorithm. I think this implies the eample does not necessarily have a technique that is cost-minimizing across all regions, as opposed to within each region. I suppose it would be good to replicate this example and produce some graphs. (I realize that computers these days provide capabalities that were not also easily available decades ago.)

Perhaps Zaffari & Sbrenna (2024) provide a model on which I should concentrate. This model seems to be in the Isard tradition, with improvements. Their modeling includes transportation costs, the spatial capacity of regions in space, and the endogenity of various variables. I can make various simplifications in developing a concrete numerical example.

Can I work through existing numerical examples in the literature? Can I find numerical values for the parameters in the model in Zaffari & Sbrenna (1984), perhaps simplified, for a reswitching example? I suppose techniques might differ in which regions, processes are run to capacity. Or, perhaps, one region specializes in manufacturing one set of commodities and the other in making manufacturing another set. I do not know how far I will get.

References

• Barnes, Trevor & Eric Sheppard. 1984. Technical choice and reswitching in space economies. Regional Science and Urban Economics 14: 345-362.
• Metcalfe, J. Stanley & Ian Steedman. 1979. Reswitching and primary input use. In Fundamental Issues in Trade Theory (ed. by I. Steedman) New York: St. Martin's Press.
• Pavlik, C. 1990. Technical reswitching: a spatial case. Environment and Planning A 22:1025-1034.
• Sheppard, Eric & Trevor Barnes. 1990. The Capitalist Space Economy: Geographical Analysis after Ricardo, Marx and Sraffa. London: Unwin Hyman.
• Zaffari, Gabriel & Giacomo Sbrenna. (2024) Sraffa goes to space: spatial elements of political economy. Review of Political Economy DOI: 10.1080/09538259.2024.2434532

'Neoclassical' Economists On The Lack Of Foundation For Some 'Neoclassical' Economics

1.0 Introduction

Last century and into this one, 'neoclassical' economists noted the lack of theoretical foundation for certain widely used models in economics. They noted that the interest rate is generally not equal to the marginal product of capital. This post quotes three prominent 'neoclassical' economics, over decades, noting the lack of theoretical foundation for such an equality.

For the purposes of this post, I have little to say about my disagreements with these authors. I will note that Sraffians have something to say about microeconomics too. I also do not want to go into here why empirical work with these unfounded models is almost always a kind of humbug.

2.0 Frank Hahn

Frank Hahn attacks my favorite school of thought. He says:

"Sraffa ... confined himself to the remark that the [missing] equation cannot be one which demands the equality of the marginal product of 'capital' and the rate of profit. ... the neoclassical economist has the same view but his reasons are not those given by Sraffa." -- Frank Hahn (1982) The neo-Ricardians. Cambridge Journal of Economics. 6(4): 362.

And again:

"The Sraffian picture of neoclassical theory is this. At any moment of time we can observe something physical called the stock of capital (K) as well as the amount of labor (L). There is a concave production function

Y = F(K, L)

where Y is output. In a neoclassical equilibrium all inputs are used and must be paid their marginal products. The latter are known once (K, L) are known. Hence the rate of profit of capital, the real wage and the distribution of income are all known once F(), K and L are known. The concavity of F further implies that the rate of return on capital is non-increasing (generally decreasing) in K. This construction, to be called the parable, Sraffians claim not to be watertight except in the single good economy. In this they are generally correct." -- Frank Hahn (1982: 370)

3.0 Edwin Burmeister

This is from a standard reference work:

"Imposing some set of conditions on the technology T() should be sufficient to ensure that the real Wicksell effect is always negative. Such conditions would be of interest - especially if they could be empirically tested - since they would validate the qualitative conclusions derived from the one-good model often used in macroeconomics without any theoretical justification for ignoring aggregation problems. Moreover, Burmeister (1977, 1979) has proved that a negative real Wicksell effect is a necessary and sufficient condition for an index of capital, k, and a neoclassical aggregate production function F(k) defined across steady-state equilibria such that (i) c = F(k), (ii) r = F’(k), and (iii) F’’(k) < 0. Unfortunately, no set of such sufficient conditions is known, but the literature on capital aggregation suggests that they would impose severe restrictions on the technology." -- Edwin Burmeister (1987). Wicksell effects. The New Palgrave.

That index is Champernowne's chain index measure of capital.

4.0 Emmanuel Farhi

Here is Emmanuel Farhi giving a lecture in 2018 agreeing with the above authors. His history of the CCC is in the first half hour. There is an accompanying paper (working paper version here). Farhi's co-author, David Rezza Baqaee, seems to be pursuing this approach.

5.0 Conclusion

Economics presents a problem for the sociology of 'knowledge'.