Elsewhere

• An applet for Marx's schemes of simple and expanded reproduction.
• Eli Cook, in The American Prospect says mainstream economists need to talk about profits.
• Simon Torracinta, in the Boston Review, decries bad (micro)economics.
• I should have mentioned Abraham Robinson and non-standard analysis in a previous post.
• Paintings by the economist Willaim Baumol.
• A painting by the economist Richard Goodwin. Apparently, he had a book.

I Was Taught That Boys Need Girls And Girls Need Boys; You Say That's Not True

I am not a biologist. In this world of 8 billion people, not all are men or women, where a man has XY chromosomes and a woman has XX chromosomes.

When fraternal twins are conceived, these two balls of cells may clump together, and one person develops. Such a human chimera may have a mixture of cells that are both XX and XY.

The SRY gene may cross over from a Y to an X chromosome. And so some men may grow up with XX chromosomes.

Klinefelter syndrome occurs in men with XXY chromosomes. Men can also have XYY or XYYY chromosomes. Women can have XXX chromosomes.

But genetics is not destiny. A long road is traversed in growing up. Sports, such as the Olympics, is about finding exceptional people who can delight us with their performances. Caster Semenya is one example, who apparently is a woman with androgen insensitivity. As I understand it, she is only one case in which the International Olympic Committee has wrong-footed itself.

In Las Salinas, in the Dominican Republic, some girls grow up to be men. Basically, some physical developments that occurred for me in the womb occur there during puberty. For some reason, this condition is more common there than elsewhere.

This post is inspired by sad current events in the United States. I have tried to concentrate above on biology. One can read Flannery O'Connor to get a Catholic sensibility on another possible complication. Deidre McCloskey is an economist who has an interesting memoir. Judith Butler supposedly is clearer in lecturing or talking about the complexities of gender than she is in her writing.

Selected References

• Judith Butler. 1990. Gender Trouble.
• Anne Fausto-Sterling. 2000. Sexing the Body: Gender Politics and the Construction of Sexuality.
• Deidre McCloskey. 1999. Crossing: A Memoir.
• Flannery O'Connor. 1955. A temple of the holy ghost.

Some Stories About Math And Science

I find certain stories of achievements in mathematics and science intriguing. In some of those I select, much that came before was overthrown. At any rate, these are stories about creations of the human mind that are tough to wrap your head around. I only claim to understand the last story.

Fermat's last theorem lacked a proof for three and a half centuries. When he first saw the theorem as a school boy, Andrew Wiles decided he was going to be a mathematican when he grew up and prove it. And he did.

I have written about the classification of finite simple groups before.

The twentieth century saw some amazing results in logic, set theory, and model theory. Gödel's incompleteness theorem, computability, the axiom of choice, the (generalized) continuum hypothesis, and the Löwenheim-Skolem theorem are very puzzling topics. Perhaps the question of the truth of the continuum hypothesis is, after last year, closer to being solved, whatever that might mean. As I understand it, both the assertion and denial of the continuum hypothesis are consistent with the axioms of Zermelo Fraenkel set theory. So its resolution would take agreement on additional axioms. Apparently, David Asperó and Ralf Schindler showed last year that one such proposed axiom implied another. I doubt I will ever understand this. I suppose perplexity at how maths mean goes back to, at least, the invention of non-Euclidean geometry.

In physics, quantum mechanics and the theory of relativity provide amazement. Their very existence is a surprise. Newtonian mechanics seemed to be the most empirically well-confirmed theory in all of science. Then, in the first couple of decades of the twentieth century, Newton was shown to be incorrect in his basic picture of the universe. At least, this is something like how Karl Popper saw it. Relativity has the surprising implication that time travel is possible in a rotating universe. Gödel showed this when he wanted to provide something for a festschrift for his friend Albert Einstein. I gather Bell's theorem shows that quantum mechanics and a limitation imposed by general relativity cannot both be right. I gather that Bell has been experimentally verified by astronomers looking at radiation passing through gravitational lenses formed from intermediate galaxies.

Political economy provides at least one story like the above. I refer to Sraffa's disproof of marginalism half a century ago.

References

• David Asperó and Ralf Schindler. 2021. MM⁺ implies (*).
• J. S. Bell. 1964. On the Einstein Podolsky Rosen paradox. Physics 1(3): 195-200.
• Stephen Budiansky. 2021. Journey to the Edge of Reason: The Life of Kurt Gödel W. W. Norton.
• Paul J. Cohen. 1963. The independence of the continuum hypothesis IProceedings of the U.S. National Academy of Sciences 50(6):1143-1148.
• Paul Cohen. 1964. The independence of the continuum hypothesis IIProceedings of the U.S. National Academy of Sciences 51(1):105-110.
• Torkel Franzen. 2005. Gödel's Theorem: An Incomplete Guide to Its Use and Abuse. Peters.
• Kurt Gödel. 1936. On formally undecidable propositions of Principia Mathematica and related systems I. Monatsheft für Mathematik und Physik 38:173-198.
• Kurt Gödel. 1938. Consistency-proof for the generalized continuum-hypothesis. Proceedings of the U.S. National Academy of Sciences 25: 220-224.
• Kurt Gödel. 1940. The consistency of the axiom of choice and the generalized continuum hypothesis with the axioms of set theory. Annals of Mathematic Studies 3.
• Kurt Gödel. 1949. An example of a new type of cosmological solutions of Einstein's field equations of gravitation. Review of Modern Physics 21: 447-450.
• Joel David Hamkins. 2011. The set-theoretic multiverse
• Morris Kline. 1982. Mathematics: The Loss of Certainty. Oxford University Press.
• Calvin Leung et al. 2018. Astronomical random numbers for quantum foundations experiments
• Edwin E. Moise. 1963. Elementary Geometry from an Advanced Standpoint. Addison-Wesley.
• Piero Sraffa. 1960. Production of Commodities by Means of Commodities: A Prelude to a Critique of Economic Theory. Cambridge University Press.
• Robert A. Wilson. 2009. The Finite Simple Groups. Springer.

The Spread Of Marxism: A Riddle

Karl Marx died on 14 March 1883. Less than 15 people attended his funeral, and Engels gave an eulogy. Marxists existed, a century later, in every country on the face of this planet, and most had political parties, some powerful, that claimed to follow Marx. How did this change from obscurity to world-wide recognition come about? What did Marx have to say that was so persuasive?

If economics were a serious subject, these questions would be explored within academic economics departments. And some universities in the United States can be taken seriously. But, as I understand it, one cannot expect mainstream economists in North America to be able to discuss these questions. One would need to be interested in economic history and the history of economics, for example, to have an informed take. Mainstream economics, I gather, are trained to deprecate such subjects. Following on the work of such economists as Donald Harris, Michio Morishima, and John Roemer, I would like those exploring Marx's economics to know some linear algebra, as well.

I suppose some might justify this incapacity and ignorance by asserting that Marx just did not have an impact on academic economics, at least in the leading schools. I am not sure this is true. Mainstream economists had to re-invent some of Marx. Consider Michal Kalecki's independent development of Keynesianism. Compare and contrast growth models, such as the Harrod-Domar and von Neumann models, with Marx's schemes of simple and expanded reproduction at the end of volume two of Capital. Employment multipliers in Leontief input-output analysis are labor values.

One can also argue the importance of Marx in the promulgation of marginalism. Eugen von Böhm-Bawerk and Philip Wicksteed explicitly argued against Marx in promoting their theories. John Bates Clark stated that his theories showed the possibility of classes living in harmony. In this sense, the erroneous doctrines that are taught today are strongly influenced by Marx, albeit in a reactionary way.

If Marx is not important to economics, why must we keep on having these purges of economics departments? Of course, those doing the purges, in their wide and deep ignorance, cannot identify a Marxist, no matter how often they look for ghosts under their bed at night.

A Theorem for Capital-Reversing

Figure 1: The Wage Frontier for a Numeric Example of a Real Wicksell Effect of Zero

Theorem: Consider a model of an economy in which n commodities are produced by means of commodities. Let Alpha be a technique in which each of the n commodities is produced by a fixed-coefficients, constant-returns-to-scale process. Suppose the Beta technique differs from Alpha only in the process operated in the nth industry. For simplicity, assume all n commodities are Sraffian basics in both techniques. Let both techniques undergo technical change, with only labor coefficients varying through time. The labor coefficients for Alpha decrease at the rate σ₁ or σ₂, while the labor coefficient for the nth industry in Beta decreases at the rate σ₂. Then the wage curves for Alpha and Beta intersect at a rate of profits of zero at time t₁ if

σ₂ t₁ = σ₁ t₁ - ln[ -z₁/z₂]

where z₁ is a linear combination of the values of the labor coefficients at time zero in the Alpha technique that decrease at rate σ₁, and z₂ is a linear combination of the remaining labor coefficients at time zero in the Alpha technique and of the labor coefficient at time zero in the Beta technique for the process producing the nth commodity.

Proof: Left as an exercise for the reader.

I consider my proof to be inelegant. This theorem is related to my previous theorem. (I've updated that post.)

Thee theorem gives an explicit condition for the wage curves for the Alpha and Beta techniques to intersect at a rate of profits of zero percent at time t₁. Suppose a switch point also exists at this time at a positive rate of profits that is less than the minimum of the maximum rate of profits for the Alpha and Beta techniques.

Around the switch point, a variation in the rate of profits or the wage is associated with no change in the quantity of labor hired per unit of net output economy as a whole.

The wage frontier illustrates for a numeric example with three produced commodities and two processes available in each industry. The techniques mentioned in the theorem are labeled "Gamma" and "Delta" in this example. Before the illustrated time in the example, this switch point is associated with a negative real Wicksell effect. Less labor is employed, per unit output of net product, at a higher wage around the switch point. After this time, it is associated with a positive real Wicksell effect. More labor is employed in the economy as a whole, given net output, at a higher wage around the switch point. The theorem gives conditions for capital-reversing to emerge, given another switch point on the frontier for the mentioned techniques.

Elsewhere

The Italian Post Keynesian Seminar on Garegnani

• The Problem with Jon Stewart interviews Stephanie Kelton and Rohan Grey.
• Samuel Fleischacker explains Adam Smith was not a propertarian.
• Jania on econophysics.
• A seminar on Stephen Marglin's Raising Keynes.

Reminder: Wages, Employment Not Determined By Supply And Demand For Labor

Figure 1: The Wage as Functions of Employment by Industry

1.0 Introduction

This post repeats a common theme of mine. It builds on an example I have previously gone on about. I use this example to graph, given the wage, the amount of labor firms would like to employ in each industry, per unit of gross output in each industry. These graphs are derived for an economy in which three commodities are produced: iron, steel, and corn. I also graph the amount of labor firms would like to employ across all industries, given that the net output of the economy consists of a unit quantity of corn. The value of this function is called an employment multiplier.

No doubt, in actual capitalist economies, some firms in some places have market power in hiring workers. Workers incur search costs in trying to find jobs whose requirements match well with their skills. Owners and managers of firms face principal agent problems. Owners, managers, workers, etc. have their own information sets at any given instant, and doubtless they are not all identitical. But, before exploring these complications, if would be nice if so many leading mainstream economists were not clueless about price theory. One might be more interested in institutions and the history of the labor movement.

2.0 Technology

Consider an economy in which three commodities, iron, steel, and corn, are produced. Two processes, as seen in Table 1 are available to produce each commodity from inputs of labor, iron, steel, and corn. Each process exhibits constant returns to scale and takes a year to produce. Each column in Table 1 specifies the inputs needed to produce a unit quantity of the commodity produced by that process. This is a model of circulating capital. All physical inputs in each process are used up in the course of the year in producing the commodity output by that process.

Table 1: The Technology

Input	Iron Industry		Steel Industry		Corn Industry
	a	b	c	d	e	f
Labor	1/3	1/10	5/2	7/20	1	3/2
Iron	1/6	2/5	1/200	1/100	1	0
Steel	1/200	1/400	1/4	3/10	0	1/4
Corn	1/300	1/300	1/300	0	0	0

A technique consists of a process in each industry. Table 2 specifies the eight techniques that can be formed from the processes specified by the technology. If you work through this example, you will find that to produce a net output of one bushel corn, inputs of iron, steel, and corn all need to be produced to reproduce the capital goods used up in producing that bushel.

Table 2: Techniques

Technique	Processes
Alpha	a, c, e
Beta	a, c, f
Gamma	a, d, e
Delta	a, d, f
Epsilon	b, c, e
Zeta	b, c, f
Eta	b, d, e
Theta	b, d, f

Each technique is represented by coefficients of production. For the Alpha technique, let a_{0, α} be a three-element row vector representing the labor coefficients, and let A_α be the 3 x 3 Leontief matrix for this technique. The first element of a_{0, α}, (1/3) person-years per ton, represents the labor input needed to produce a ton of iron. The first column of A_α represents the inputs of iron, steel, and corn needed to produce a ton of iron. A parallel notation is used for the other seven techniques.

Suppose the net output of the economy is a bushel corn. A bushel corn is also the numeraire.

3.0 The Price System

Prices of production are defined to be constant spot prices that allow the smooth reproduction of the economy. Suppose Alpha is the cost-minimizing technique. Let p be the three-element row matrix designating the prices of iron, steel, and corn. I make the assumption that markets are such that the rate of profits in the iron, steel, and corn industries are (r s₁), (r s₂), and (r s₃), respectively. Suppose S is a diagonal matrix with the obvious elements along the diagonal, and I designates the identity matrix. Then prices of production satisfy the following system of equations:

p_α A_α (I + r S) + w_α a_{0, α} = p_α

I choose a bushel of corn to be the numeraire. If e₃ is the last column of the identity matrix, the following equation specifies the numeraire:

p_α e₃ = 1

As is not surprising, the above system of equations has one degree of freedom. One can solve for the wage, w_α(r), as a function of the scale factor for the rate of profits, r. The wage curve is a downward-sloping curve that intercepts both the axis for the wage and the scale factor at positive values. A similar function can be derived the other techniques, and they can be graphed in the same diagram.

4.0 The Choice of Technique

Figure 2 graphs the wage curves for the techniques that are cost-minimizing for some feasible wage, given markups by industry. The outer envelope is the wage frontier. The cost-minimizing technique at a given wage is the technique with the right-most wage curve at that wage. The cost-minimizing techniques at each wage and the switch points between techniques are noted on the figure.

Figure 2: The Wage Frontier

5.0 Wages and Employment

For each technique, one can calculate the employment required across all three industries to produce a net product of a bushel corn. In these calculations, the processes in a technique are operated at a level so as to replace the iron, steel, and corn used up in producing that bushel of corn. Since which technique is cost-minimizing at a given wage is shown above, one can plot the wage against employment, as in Figure 3. In some sense, this is a macroeconomic labor demand function. On the other hand, if one does not get well-behaved supply and demand functions for labor, one might want to say that supply and demand does not apply here. Notice the switch point between the Gamma and Delta techniques. Around this switch point, a higher wage is associated with firms wanting to employ more workers.

Figure 3: The Wage as a Function of Employment Across Industries

The labor coefficient in each industry is specified along with each technique. Figure 1, at the top of this post, graphs employment in each industry per unit gross product. Here, a higher wage around the switch point between the Gamma and Delta techniques is associated with firms wanting to employ more labor per bushel corn produced as gross output in the corn industry. This reverse substitution of labor can occur around a switch point in which capital-reversing does not occur and vice versa.

6.0 The Effects of Markups

In the above story, the markup in the steel industry is less than the markups in the iron and corn industries. One might think of this as a deviation from competitive markets. In this conception, markets are competitive when markups are unity in all industries.

Figure 4 illustrates how the sequence of techniques along the wage frontier varies with the markup in the steel industry. The result of the specific markups used above is that the Beta technique is cost-minimizing at a low enough wage. That is the second process in the corn-producing industry recurs. The first corn-producing process also recurs.

Figure 4: The Variation of the Wage Frontier with the Markup in the Steel Industry

If those investing in the iron and corn industries are able to persistently impose even greater barriers to entry, the markup in the steel industry would be even lower. Evenually, the Alpha and the Gamma techniques would not be cost-minimizing at any wage. Neither process in the corn industry would recur. The instance of capital-reversing would also be destroyed. The same follows if the markup in the steel industry exceeds the markups in the iron and corn industry sufficiently.

7.0 Conclusion

As far as I know, mainstream economists have been teaching what has been known to be, at best, incorrect for half a century. Are they fools or knaves? What accounts for this extraordinary intellectual bankruptcy?