## Tuesday, December 27, 2022

### A Fluke Case For Extensive Rent

 Figure 1: Wage Curves and Rent for an Example of Extensive Rent

This is basically an introduction to a draft research article. Maybe I have difficulty in justifying paying attention to the type of fluke points I have been exploring and in formally defining why they are flukes.

In this post, I present a partition of a parameter space associated with an example of extensive rent. It is probably too complicated to replace this example. Anyways, I have constructed a new fluke case. Here a switch point associated with a variation in the order of fertility is also associated with a variation in the order of rentability. I suppose one can read David Ricardo as mistakenly thinking this was the general case.

The analysis of the choice of technique in models of extensive rent can be based on the construction of wage curves, even though the outer envelope does not represent the cost-minimizing technique. The orders of fertility and rentability are emphasized here. The order of fertility is defined for specified techniques, in which a single quality of land is used in each technique and that land pays no rent. At a given rate of profits, the qualities of land are ordered by wages, with the most fertile land paying the highest wage. The order of rentability specifies the sequence of different qualities of lands from high rent per acre to low rent per acre. Both orders may vary with the wage or the rate of profits. Table 1 presents coefficients of production for an example.

 Input Iron Industry Corn Industry I II III IV Labor 1 a0,2 91/250 67/100 Type 1 Land 0 49/100 0 0 Type 2 Land 0 0 59/100 0 Type 3 Land 0 0 0 9/20 Iron 9/20 a1,2 9/10000 67/1000 Corn 2 6/125 27/100 3/20

How much corn can be produced is constrained by the available quantities of each type of land. Endowments of land and requirements for use must be among the givens to analyze the choice of technique in this example. Suppose one hundred acres of each type of land are available, and net output is such that all three types of land must be farmed. One type will be only partially farmed. The iron-producing process must be operated in each of the three economically viable techniques. Table 2 describes which type of lands are fully cultivated and which type of land is left partially fallow in each of the Alpha, Beta, and Gamma techniques.

 Technique Land Type 1 Type 2 Type 3 Alpha Fully farmed Fully farmed Partially farmed Beta Partially farmed Fully farmed Fully farmed Gamma Fully farmed Partially farmed Fully farmed

For a given technique, the rent on a type of land only partially farmed is zero since it is not scarce. The wage curves in the left pane in Figure 1 are constructed, for each technique, from the price equations provided by the iron-producing process and the corn-producing process for the land that pays no rent. The choice of technique depends on both income distribution and requirements for use (Quadrio-Curzio 1980). Suppose the rate of profits is taken as given. Then the r–order of efficiency or fertility is the order of the wage curves downwards, until requirements for use are satisfied. Since all three types of land must be somewhat cultivated in the example, the wage frontier is the inner envelope of the wage curves. For the illustrated parameters, Alpha is cost-minimizing, and the order of fertility between the switch points Type 2, Type 1, and Type 3 lands. At the switch point on the wage axis, Type 1 and Type 2 lands are equally fertile. For rates of profits exceeding that at the second switch point, the order of fertility is Type 2, Type 3, and Type 1 lands.

One can calculate the cost of capital goods at the given rate of profits and the cost of labor inputs for corn-producing processes on each land that pays a rent. Since coefficients of production are specified per bushel corn produced, the revenues from each of these processes, at a unit level, are the same as the process operated on the least fertile land, which pays no rent. Rent is the difference between revenues and non-land costs on scarce lands. Rent per acre is plotted in the right pane for Figure 1. The order of rentability is the order of lands by rent per acre. The order of rentability is the same as the order of fertility for the given parameters.

The example in Figure 1 is a fluke case at a rate of profits of zero is a fluke case three times over. A switch point on the outer frontier lies on the wage axis. The rent per acre for Type 1 and Type 2 lands is the same at a rate of profits of zero. And the order of efficiency and the order of rentability both change, in some sense, at a rate of profits of zero.

 Figure 2: A Portion of the Parameter Space

To the northeast in Figure 2 (Region 1), the switch point at a rate of profits of zero has disappeared over the wage axis. The order of fertility matches the order of rentability, even with the variation in the choice of technique with distribution. To the northwest (Region 2), this switch point appears at a positive rate of profits, and the order of fertility varies with the rate of profits where the Alpha technique is cost-minimizing. The point of intersection on the right-hand pane in Figure 1 still occurs at a rate of profits of zero. Towards the southwest (Region 3), this intersection occurs at a positive rate of profits, but still at a smaller rate of profits than the switch point for which the order of fertility varies for the Alpha technique. The order of fertility and the order of rentability are the same at a rate of profits of zero, but reversed from what they are in Region 1. Region 3 and Region 4 differ in that the switch point for which the order of fertility varies for the Alpha technique occurs at a rate of profits less than the rate of profits at which the rent per acre for Type 1 and Type 2 lands are equal. Finally, the range of profits for which the rate of fertility differs, for the Alpha tecnique, from what it is in Region 1 has vanished. Table 3 summarizes this discussion.

 Region Range LandPartiallyFarmed Order ofFertility Order ofRentability 1 0 ≤ r ≤ r1 Type 3 Type 2 > Type 1 Type 2 > Type 1 r1 ≤ r ≤ Rβ Type 1 Type 2 > Type 3 Type 2 > Type 3 2 0 ≤ r ≤ r1 Type 3 Type 1 > Type 2 Type 2 > Type 1 r1 ≤ r ≤ r2 Type 2 > Type 1 r2 ≤ r ≤ Rβ Type 1 Type 2 > Type 3 Type 2 > Type 3 3 0 ≤ r ≤ r1 Type 3 Type 1 > Type 2 Type 1 > Type 2 r1 ≤ r ≤ r2 Type 2 > Type 1 r2 ≤ r ≤ r3 Type 2 > Type 1 r3 ≤ r ≤ Rβ Type 1 Type 2 > Type 3 Type 2 > Type 3 4 0 ≤ r ≤ r1 Type 3 Type 1 > Type 2 Type 1 > Type 2 r1 ≤ r ≤ r2 Type 2 > Type 1 r2 ≤ r ≤ r3 Type 2 > Type 1 r3 ≤ r ≤ Rβ Type 1 Type 2 > Type 3 Type 2 > Type 3 5 0 ≤ r ≤ r1 Type 3 Type 2 > Type 1 Type 1 > Type 2 r1 ≤ r ≤ r2 Type 2 > Type 1 r2 ≤ r ≤ Rβ Type 1 Type 2 > Type 3 Type 2 > Type 3

The correct analysis of extensive rent does not conform to reasoning based on supply and demand. If you disagree, can you tell a supply and demand story for this example? I find it difficult to directly confront these two modes of reasoning, one correct and one known, for decades, to be confused. I also want to say that partitioning parameter spaces with fluke switch points can illuminate the impact of technical change on income dynamics. A movement from the northeast to the southwest in Figure 2 is an improvement in technology. Such an improvement has an impact on how landlords view struggles between capitalists and workers over the rate of profits and wages. But I am still putting aside the difficult question of the supposed tendency of market prices to approach prices of production.

## Saturday, December 24, 2022

### Elsewhere

 Mary Filippo's "My Mis-Education in 3 Graphics"

## Wednesday, December 21, 2022

### An Outline Of A History Of Socialism

In my study of economics, I have learned a bit about socialism.

Writing a book based on this outline is a years-long project. Some parts are not filled out in the outline because I know too much and my thoughts are unorganized (not that you might disagree with my emphasis and story). Others are not filled out because I know too little. I am aware I have spelling mistakes. Some needs to be reorganized.

• Introduction, Overall Themes
• Socialists advocate that capitalism be replaced by a post capitalist society in which the workers, who constitute the vast majority of the population in an industrial society, collectively make decisions on what to produce and how to produce it. From this perspective, advocates of socialism cannot exist until after capitalism has been established. Furthermore, one cannot advocate for an industrial proletariat prior to the existence of a working class. Thus, a history of socialism should properly start around the time of, say, the French revolution. (Alexander Gray disagrees)
• Hardly anybody at Marx's funeral. A century later political parties in every country claimed to be followers of Marx, in some sense. Communists in power ruled how much? of the world's population. Social democracy and democratic socialism dominant in Western Europe.
• When political parties calling themselves labor or social democratic emerged in the 19th century, socialism (or social democracy?) and communism were not sharply distinguished, although they were distinguished from liberalism. (caveat: J. S. Mill). They were distinguished in the 20th century.
• With the new left and post 1989, other radical traditions became more prominent (Greens, feminists, workerism in Italy, anarchists(?), anti-imperialist movements in third world(?), black power…)
• When parties attain some measure of government responsibility (in parliament, as part of coalition governments, as ruling party (including the USSR)), they make compromises. More radical movements arise, to be later co-opted and to become non-pure. The cycle repeats. Examples?
• What to say about liberalism, fascism, neo-liberalism, general background? Reference Hobsbawm four volume series.
• C. B. Macpherson's idea that the 18th, 19th, and 20th centuries were about civil, political, and economics rights, respectively. Abolition of slavery, extensive of franchise, right to form labor unions. Socialism illegal much of this time, with newspapers published by exiles and conferences outside home country. Socialists elected when legal.
• This history is Eurocentric. Tries not to be a history of ideas, but more a summary of political institutions and political actors.
• Not covering Hegel much. Contrast with Kolakowski.
• Not claiming to be very original. Trying to be terse.
• Range of parties and movements called socialist, etc. Strength is that if you reject one tendency, there is always another.
• Precursors
• The bible, old and new testaments. (what about other religions? E.g. Confucious, Analects, wants social harmony. Constraints on rulers?)
• Utopias: Plato's Republic, Thomas More's Utopia, Campanella's The City of the Sun, Swift's Gulliver’s Travels? Extend to 19th and 20th century?
• Peasant revolts (jacqueries): French peasants' revolt of 1358, John Ball and 1381 peasants' revolt, 16th century peasants' revolt in Germany (1525, Engels' commentary)
• Millenarianism
• Anabaptists, Diggers, Levellers, Muggletonians, the New Model Army, Ranters, and Quakers in the English revolution
• Political philosophy: Augustine's City of God, Machiavelli, Hobbes, Locke, Rousseau (Man is born free, but everywhere in chains. The first man who, having enclosed a piece of ground, bethought himself of saying, "This is mine", and found people simple enough to believe him, was the real founder of civil society.). What is Republicanism? Thomas Paine? William Godwin?
• Enlightenment. liberalism is progressive and an ideology for the rising bourgeois
• American revolution. According to Hannah Arendt, not a social revolution. Beards’ idea that the constitutional convention was a counter revolution.
• The French Revolution
• Socialists Before Marx
• France: Saint Simon, Fourier, Proudhon
• Ricardian socialists
• Chartism
• Germany and revolutions of 1848
• Marx and Engels
"German philosophy, English political economy and French socialism… these three sources of Marxism, … are also its component parts..." (Lenin 1977)
• Disagreements Around the Time of the First International
• Distinctions among different kinds of communists in the Communist Manifesto
• Utopian or scientific socialism in Engel's pamphlet. Marx's involvement. Reprint of chapters in Anti-During.
• No definitive distinction between socialism and communism. Maybe communism was perceived as more radical
• Marx's letter, critique of the Gotha program. The Gotha program was put forth at the Gotha conference (1875). The German social democratic party was formed by a compromise between followers of Lasalle and Liebknecht. Socialism is the first stage after capitalism, communism the higher stage. Lenin says same in State and Revolution.
• Communards, Paris commune. (1871). Blanqui more involved than Marx. Nickname for Marx in British newspapers, something like the red doctor. Dictatorship of the proletariat.
• Michel Bakunin anarchist who said Marx would lead to a dictatorship. The end of the first international
• The Second International Through World War I
• The socialist or second international. parliamentary parties. The road to socialism through democratic means. Appreciative of bourgeois virtues like freedom of the press.
• The Erfurt program (1891) of Geman social democratic party was Marxist
• Bernstein and revisionism. 1899?. The final destination is nothing, the journey is everything. Split between Mensheviks and Bolsheviks. What is to be Done? Split in France? Sorel more leftist, Jaures to the right. Compare support for Bernstein from labor unions with Lenin’s critique of economism.
• Fabian socialism in Britain.
• Socialist in parliaments felt obligation to vote for military budgets.
• At Stuttgart, in 1907, the Second International adopted a resolution, declaring (later conferences, similar resolutions):
"Should war break out despite everything, it is the duty of Socialists to act for its rapid conclusion and to work with all their strength to utilize the economic and political crisis provoked by the war to rouse the peoples and thus accelerate the abolition of the rule of the capitalist class." (Mandel 1978)
• Second and a half international. Zimmerwald Conference. Political prisoners.
• Spartacus revolution. Murder of Karl Liebknecht and Rosa Luxemburg
• The distinction between socialism and communism
• Should social democracy be a separate thing? Not trying to replace capitalism, but moderate its rough divisions.
• Other divisions: Council communism. Anarcho-syndicalism. Sorel. POUM in Spain. Gramsci in Turin. Lenin denounced some of this in Left Wing Communism: An Infantile Delusion. Rosa Luxemburg. Twists and turns of Bolsheviks? Divisions among followers of Trotsky. Later: Eurocommunism.
• Theoretical developments. Finance capital. Theories of imperialism. Hobson, Luxemburg, Bukharin, Lenin
• The October (Bolshevik) revolution and the Russian civil war
• Georgi Plekhanov and Lenin were leaders of the Russian social democratic party.
• Russian revolutions: 1905, February and October. Explain what is a soviet, Hannah Arendt. Lenin's slogan was all power to the soviets, but immediately took power away from them. What is a commissar?
• What is to be done? Economism: improving working conditions, more pay, shorter hours. Need an outside vanguard to have something more than working class consciousness.
• Stalinism
• Josef Stalin had consolidated power in the USSR, first in alliance with Nikolai Bukharin, Grigory Zinoviev and Lev Kamenev against Leon Trotsky and Evgeny Preobrazhensky in 1923. Vladimir Lenin died in January 1924. Stalin and Bukharin then defeated Zinoviev and Kamenev in 1924. In 1927, Stalin and Bukharin overcame the United Opposition of Trotsky, Zinoviev, and Kamenev. Finally, Bukharin was stripped of power in 1928, leaving Stalin ruling. Trotsky was exiled in 1929 and assassinated in Mexico in 1940. Preobrazhensky was executed in 1937 during the great purge, Zinoviev and Kamenev were tried and killed in 1936. Bukharin was shot after the most famous Moscow show trial in 1938.
• New Economic Policy (NEP) over Preobrazhensky’s primitive socialist accumulation
• Socialism in one country over Trotsky's permanent revolution.
• Other communist parties are to be subordinated to the CPSU
• Collectivization of agriculture when NEP was supposedly not working
• Need to say something more about purges, thaws, show trials.
• World War II. Social democracy as social fascists. No united front. Alliance with Hitler, and then otherwise.
• Selection among Stalin’s writings. History of the communist party. Comments on a text book on economics. (So much for Preobrazhensky and Bukharin’s ABCs.) Have we achieved communism yet? Summary of dialectics.
• Social Democracy in Europe
• Revolutions in Hungary, Germany
• FIAT occupation of factories (this was not a sit-down strike, like UAW organizing. The workers ran the factories and produced cars.) Italian factory councils
• General Strike in Great Britain
• Red Vienna
• Debts and reparation payments from World War I bedeviled many European countries, with the attendant inflation. Cite Keynes' tract
• Great depression
• Socialists did not do well in responding. Gunnar Myrdal and Kalecki inventing Keynesianism themselves.
• Fascism and World War II
• Fascism was on the rise, with Mussolini having consolidated power in Italy around 1925, after elections in April 1924. The fascists enacted laws against association, removing Mussolini’s responsibility to parliament, giving him power to enact laws independently of the Chamber of Deputies, outlawing the right to strike, and outlawing all political parties except the National Fascist Party (PNF) (Fretigne 2021).
• POUM and Spanish civil war. Cite Orwell’s Homage to Catalonia, Chomsky? Auden’s Spain. But today the struggle
• After Stalin
• The Warsaw Pact
• Khrushchev’s secret speech
• Mao and China
• Tito, Yugoslavia and Milovan Djilas's The New Class
• Hungary 1956
• Czechoslovakia and the Prague Spring
• I ought to say something about French, Italian communist parties where they are important political players
• From the End of World War II to the New Left
• The golden age, Les Trente Glorieuses. The so-called crisis of democracy and neoliberal reaction
• Bretton Woods
• The U.N., the IMF, the World Bank, WTO/GATT
• Universal Declaration of Human Rights
• The Bandung Conference (1955), the third world, non-aligned countries
• Anticolonialism. Fanon. Amin. Theories of unequal exchange.
• The Frankfurt School (Adorno, Horkheimer, Benjamin, Fromm, Marcuse)
• Gramsci, Lukas, Althusser
• Paul Sweezy and Paul Baran
• Notice theorists are no longer leaders of political parties. (Exception: Alain Lipietz)
• Norman Thomas
• McCarthyism
• The New Left
• The Port Huron statement. SDS. Michael Harrington. Looking forward to DSA
• Cuba. Other third world revolutionaries.
• Civil Rights and Martin Luther King
• Vietnam
• Paris 1968, the 68ers, Danny the Red. Be realistic, demand the impossible. Under the paving stones, the beach.
• Lots more about 1968. What can a poor boy do except sing for a rock and roll band? (I saw Bruce do this, and I'm sure he knows this song was in response to Paris.)
• Waves of feminism
• Stonewall
• Prospect of liberation in many parts of the world. Violence and sadness and the security state consciously undermining all.
• After 1989
• Emphasize that 1989 was more than overthrow of actually existing socialism: Philippines people power, Chile, South Africa (maybe write about Biko). Hopeful at the time.
• 1789, 1848, 1871, 1917, 1968, 1989
• Failures of “big bangs”
• Communist parties changing their names
• Occupy wall street. David Greider
• 2008 Global financial crisis
• Covid depression
• Conclusion

## Monday, December 19, 2022

### Journals To Which I Might Submit Articles

I am thinking about my work of fluke switch points. Here are some possibilities, some of which are more of a stretch:

Cato the elder ended every speech with, "Carthago delenda est", whatever the topic. Maybe articles in these journals should end with "marginalism was destroyed a half-century ago".

Somebody, maybe not me, should be submitting recaps of why marginalism is mistaken to outlets like the Economic Journal or the American Economic Review. I fear mainstream economists are still ignorant of results in price theory established half a century ago and reduced to textbook teaching at least half a century ago.

## Wednesday, December 14, 2022

### Marx's Theory Of Value

 Victor Margariño explains labor theory of value to Vaush

Marx sets his theory of value within the capitalist (or bourgeois) mode of production:

"The wealth of those societies in which the capitalist mode of production prevails, presents itself as 'an immense accumulation of commodities'..." (Marx 2010, first sentence of chapter 1)

Feudal societies, with lords and serfs, and classical societies, such as the Roman empire with its slaves, present other modes of production. Although this exposition starts from the same point as Marx, it deviates from his dialectical method of presentation.

In the capitalist mode of production, workers sell their ability to labor under the direction of others, that is, their labor power, to the owners of firms for money. The workers use this money to buy the goods and services they need to survive. (As emphasized by feminists, the sustaining of the workers includes much care work and household production outside the world of commodity production.) The owners of firms, that is, the capitalists, own the equipment, raw material, and semi-finished goods that the workers use to manufacture these goods and services. They have purchased these means of production on various markets. After the workers have used the means of production to produce a commodity, the capitalists also own the product. The capitalists sell the product of the labor of the workers on the market for money. A commodity is produced for the market, to be used by others. To be bought and sold, it must have a use value, some attributes that make it useful for others. Buying and selling occurs regularly and repetitively under capitalism, not only at the edges of society.

These characteristics of the capitalist mode of production are distinctive. Slaves and serfs are not free to sell their labor power on their own account. Labor power is also not sold by self-employed artisans. Peasants living in a pre-capitalist communal village (for example, the Russian mir) do not produce commodities to be sold to their neighbors. The slaves and serfs producing the goods needed to sustain the owner's estate are not producing commodities.

In capitalist societies, most workers are not producing commodities to satisfy final demand. Rather, they are producing commodities that are bought and sold in a vast network among firms. I like to characterize this network by Leontief input-output matrices in physical terms. National income and product accounts (NIPA) are usually presented in terms of monetary flows. A Leontief matrix shows farmers purchasing tractors of specified kinds, fertilizers, and so on from other industries. The seeds are purchased from agriculture itself. The manufacturers of tractors buy steel, and the steel industry buys iron. For a self-sustaining economy, these quantity flows must be constantly repeated. Even while steel manufacturers are running down stocks of iron in producing the current output, they are also replenishing these stocks with current purchases of iron.

The production of final demand with a given technology implies the distribution of the employed labor force of a nation among industries. For each commodity, how much additional labor would be employed if the net output were increased by one unit of that commodity? In Leontief analysis, the answer to this question yields employment multipliers. An employment multiplier is the labor directly and indirectly employed to produce one additional unit of that commodity in s self-sustaining way. It is also known as the labor value of the commodity. Notice that this labor time will be distributed over many industries. Steel production requires mining of iron, the generation of electricity to operate the mills and mines, the operation of railroads and trucking to ship iron ore to the mills and the steel to where it is needed, and so on. Labor values reflect social relationships between workers not immediately apparent in the buying and selling of commodities.

I now want to point to some more advanced mathematics, easiest to set out when no joint production exists and when the production of each commodity requires inputs, either directly or indirectly, of all other commodities. Sraffa (1960) goes to great lengths, not always successfully, to argue that this approach works with the relaxation of these assumptions. Suppose the Leontief matrix shows the production of n commodities, and that the technique represented by this matrix allows for the production of a surplus product, after the replacement of the means of production used up in satisfying final demand. Then the Leontief matrix has n eigenvalues. Consider the maximum eigenvalue, which is strictly positive and less than unity. The components of the corresponding eigenvector are all strictly positive. With appropriate scaling, this eigenvector is Sraffa's standard commodity.

The standard commodity represents the surplus produced over the course of, say, a year. It is a composite commodity. The means of production used in the production of the standard commodity are in the same ratios as the standard commodity. Proportions of the standard commodity map directly to proportions of the employed labor force. If final demand was the standard commodity, production would expand each of the means of production at the same rate, a rate related to the maximum eigenvalue of the Leontief matrix. If all of final demand were invested, the economy would expand along the von Neumann (1945-1946) growth path.

Given the wage, one can find prices of (re)production associated with Leontief matrices. For a competitive economy, the same rate of profits is obtained in each industry. Inputs are evaluated at prices of production, as are outputs. For any wage not exceeding the maximum wage, prices of production are strictly positive and well-defined, given the technique. In effect, prices of production show an outcome that validates managers of firms in their decisions about which processes to operate in each industry and the levels at which these processes are operated.

Market prices can be expected to deviate from prices of production. The possibility of such deviations is known as the realization problem. These deviations can be expected to result in disinvestment in some industries and expanded investment in other industries. Furthermore, the level of effective demand may lead to the overutilization or underutilization of capacity. Likewise, the level of employment consistent with effective demand need not match the size of the labor force looking for work. Capacity utilization and unemployment are separate questions; relative prices cannot be expected to vary so as to solve both problems simultaneously. Furthermore, decisions that are validated by prices of production at one moment of time cannot be expected to be validated at another time. The size and composition of final demand and technology are continually changing, not completely exogenously.

A simple labor theory of value asserts that prices of production are (proportional to) labor values. Marx explicitly rejects that theory:

"If prices actually differ from values, we must, first of all, reduce the former to the latter, in other words, treat the difference as accidental in order that the phenomena may be observed in their purity, and our observations not interfered with by disturbing circumstances that have nothing to do with the process in question. We know, moreover, that this reduction is no mere scientific process. The continual oscillations in prices, their rising and falling, compensate each other, and reduce themselves to an average price, which is their hidden regulator. It forms the guiding star of the merchant or the manufacturer in every undertaking that requires time. He knows that when a long period of time is taken, commodities are sold neither over nor under, but at their average price. If therefore he thought about the matter at all, he would formulate the problem of the formation of capital as follows: How can we account for the origin of capital on the supposition that prices are regulated by the average price, i. e., ultimately by the value of the commodities? I say 'ultimately', because average prices do not directly coincide with the values of commodities, as Adam Smith, Ricardo, and others believe." (Marx 2010, last footnote in chapter 5)

And again:

"The calculations given in the text are intended merely as illustrations. We have in fact assumed that price = values. We shall, however, see, in Book III, that even in the case of average prices the assumption cannot be made in this very simple manner." (Marx 2010, last footnote in chapter 9, section 1)

Marx adopts a simple labor theory of value in the first volume of Capital for illustration and for methodological reasons in explaining the origin of returns to capital. But he knows the theory is false.

The relationship between labor values and prices is particularly transparent in the production of the standard commodity, assuming single production. Accordingly, assume that the net output of the economy is the standard commodity and that the standard commodity is the numeraire in which wages are denominated. Then the value of the means of production, C, is the same, whether calculated in labor values or prices of production. Likewise, net output, V + S, is the same, whether calculated in labor values or prices of production. These aggregate quantities are independent of how net output is divided among payments to workers, V, and returns to capital, S. The rate of profits in the system of prices of production is the same as the rate of profits in the system of labor values:

r = S/(C + V) = (S/V)/((C/V) + 1) = e/(1 + occ)

where e is the rate of exploitation, also known as the rate of surplus value, and occ is the organic composition of capital. I believe these invariants solve the notorious so-called transformation problem. They match Marx's assertions in volume 3 of Capital.

Marx, of course, does not write about Leontief matrices and their eigenvalues and eigenvectors. The analysis of both Leontief and Marx, however, views production as a circular process in which commodities are produced, with labor, by means of commodities. Furthermore, Marx does write about a commodity of average organic composition. And Sraffa’s standard commodity is exactly that commodity.

Sraffa's theory of prices of production is an open model, with a degree of freedom for the functional distribution of income. If the wage or the rate of profits is taken as given, the model is closed. Other closures are possible. Formally, one could assume households are intertemporal utility maximizers (Marglin 1984). In this sense, Marx's theory of value is consistent with marginalist economics. This closure could provide multiple equilibria. Given final demand and technology, the quantities of capital goods used as inputs are solved by the model. They are not data, and the marginalist revolution was, at best, mistaken on a fundamental level. Political economy is not an analysis of the allocation of scarce resources among alternative uses. Rather, it includes an analysis of the conditions needed for the material reproduction of capitalist societies.

References
• Bukharin, Nikolai. 1972. Economic Theory of the Leisure Class. New York: Monthly Review Press.
• Leontief, Wassily W. 1936. Quantitative Input and Output Relations in the Economic Systems of the United States. Review of Economic Statistics 18 (3): 195-125.
• Marglin, Stephen A. 1984. Growth, Distribution, and Prices. Cambridge: Harvard University Press.
• Marx, Karl. 2010. Capital: A Critique of Political Economy, Volume 1. Marx-Engels Collected Works V. 35. Lawrence & Wishart.
• Sraffa, Piero. 1960. Production of Commodities by Means of Commodities: A Prelude to a Critique of Economic Theory. Cambridge: Cambridge University Press.
• von Neumann, John. 1945-1946. A model of general economic equilibrium. Review of Economic Studies 13 (1): 1-9.

## Friday, December 09, 2022

### Fluke Switch Points

Exploring perturbations of four examples of fluke switch points provides a brief survey of some aspects of prices of production. The examples arise in, respectively, models of circulating capital, fixed capital, extensive rent, and intensive rent. The reverse substitution of labor, reswitching, and capital reversing, for example, are contrasted with genuine fluke cases.

These posts present examples of fluke switch points. Each example is of a fluke case in at least two ways. Either two switch points are flukes, or a switch point exhibits two fluke properties. For example, one fluke switch point might be on the axis for the rate of profits and another might be on the wage axis. I say 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.

This analysis builds on post-Sraffian price theory. Kurz & Salvadori (1995) provide a comprehensive textbook treatment of the analysis of prices of production and of the choice of technique, including in models with circulating capital, fixed capital, and extensive and intensive rent. Parameter spaces for fluke switch points are explored by Vienneau (2021 and 2022). Prices of production, in competitive markets, are such that operated processes can satisfy requirements for use after replacing commodity inputs required for production. A single rate of profits is made in all operated processes, and no extra profits are obtainable in non-operated processes.

National income and product accounts (NIPAs) provide Leontief matrices with elements expressed in price ratios. Given price indices by industry or normalization methods, one can obtain Leontief matrices in physical terms. These matrices, with certain abstractions, can be used to find prices of production corresponding to any functional distribution of income. Price flows include vintage technologies with old plants earning quasi rents. Fixed capital is hard to handle rigorously in empirical data (Kurz 2021). The rate of profits cannot be expected to be uniform across industries, since competitive conditions do not always prevail. Be that as it may, a problem of the choice of technique arises by combining Leontief matrices across time or countries. Han & Schefold (2006) and Zambelli (2018), following roughly the above approach, find some examples, albeit not many, of reverse labor substitution, of the reswitching of techniques, and of capital reversing.

Given the knife edge property of the fluke cases presented in the article, they cannot be expected to be observed in the empirical data. But local perturbations of fluke cases point to the possibility of qualitative differences in the analysis of the choice of technique. Exploring parameter spaces near selected fluke cases facilitates a brief survey of some aspects of prices of production.

These posts do not explore how such qualitative change impacts the temporal dynamics of market prices, an unsolved problem in price theory (Kurz & Salvadori 2022, Bellino 2011). If one accepts that prices of production, given technology, net output, and a distributive variable, provide information on tendencies in market prices, perturbations of coefficients of production indicate qualitative changes in such tendencies. In examples or reverse labor substitution and of capital reversing, higher wages are associated with the adoption of a technique in which more labor is hired, respectively, per unit gross output in a given industry and per unit net output in the economy overall. But variations in coefficients of production may eliminate such possibilities. Perturbing fluke cases in numerical examples suggests which specific qualitative changes are possible.

These posts also do not draw conclusions about the probability of, for example, capital-reversing arising in actual data. Fluke cases correspond to lower-dimensional manifolds in multi-dimensional parameter spaces. Regions in the parameter space corresponding to reswitching, for example, are of strictly positive measure. I am reluctant to draw any conclusions from the relative sizes, in these examples, of regions formed by partitions from fluke cases. Schefold (2022) notes that given random matrices of coefficients of production, reswitching is rare. Very few techniques, however, lie on the wage-rate of profits frontier. His argument critiquing marginalist economics along these lines seems independent of the approach in these posts.

This post is about four other posts. The first explores an example of two fluke switch points in a model of circulating capital. The reverse substitution of labor results from a perturbation of this fluke case. The second post presents an example of a fluke switch point in a model of fixed capital. Reswitching emerges with perturbations of selected parameters The third post describes two fluke switch points in a model of extensive rent. The order of fertility differs from the order of rentability with appropriate perturbations. The fourth post presents a fluke switch point in a model of intensive rent and discusses the emergence of the non-existence and non-uniqueness of a cost-minimizing technique.

Many more fluke cases, some of different types, can be constructed for those with the requisite patience. The reverse substitution of labor, reswitching, capital reversing, the association of the truncation of the economic life of a machine with the choice of a more capital-intensive technique, a divergence between the order of fertility and the order of rentability, the variation in the existence of rent with the rate of profits, and the non-uniqueness and the non-existence of a cost-minimizing technique are not fluke cases. This article demonstrates this result by contrasting these possibilities with genuine fluke cases.

## Thursday, December 01, 2022

### The Emergence Of Multiple Cost-Minimizing Techniques

 Figure 1: Wage Curves and Rent for an Example of Intensive Rent

This post is a rewrite of this.

The analysis of the choice of technique, in my previous three posts in this series, has always been based on the construction of a wage-rate of profits frontier. Given a technology in which requirements for use can be satisfied, prices of production for a feasible technique, including the wage, are uniquely determined by the given rate of profits. If the rate of profits is in a range where such prices are non-negative for at least one technique, one of the techniques is uniquely cost-minimizing, except at switch points. These properties do not necessarily hold in models of general joint production. An examination of local perturbations in an example of intensive rent illustrates surprising possibilities.

Table 1 presents coefficients of production, a perturbation of an example from D'Agata (1983). Only one type of land exists, and three processes are known for producing corn on it. The scarcity of land is shown by the possibility of two corn-producing processes being operated side-by-side in the cost-minimizing technique.

 Input Industry Iron Steel Corn I II III IV V Labor 1 1 1 57/20 21/20 Land 0 0 1 13/10 21/20 Iron 0 0 1/10 a1,4 a1,5 Steel 0 0 2/5 13/100 21/200 Corn 1/10 3/5 1/10 2/5 21/50

Following D'Agata, assume that one hundred acres of land are available and that net output consists of 90 tons iron, 60 tons steel, and 19 bushels corn. The net output is also the numeraire. All three commodities must be produced for any composition of net output. Table 2 lists the available techniques. Only Alpha, Delta, and Epsilon are feasible for the parameter ranges considered. Not all land is farmed and only one corn-producing process is operated under Alpha. Two corn-producing processes are operated together under Delta and Epsilon.

 Technique Processes Alpha I, II, III Beta I, II, IV Gamma I, II, V Delta I, II, III, IV Epsilon I, II, III, V Zeta I, II, IV, V

Figure 1 shows the wage curves for feasible techniques at a selected parametrization. I take the wage curve for a technique to be defined only for non-negative rates of profits at which the wage, rent per acre, and the prices of produced commodities are non-negative. The wage is negative for Delta for rates of profits below that at the switch point, and rent is negative for rates of profits greater than that at the switch point. Rent is negative for Epsilon for rates of profits less than at the switch point, and the wage is negative for greater rates of profits. Thus, the switch point is the only point on the wage curves for Delta and Epsilon. The switch point is a fluke in at least two ways. It is a switch point for three techniques, not two. And it is on the axis for the rate of profits.

Figure 2 shows a partition of the parameter space around this fluke case. An intersection of three wage curves over the axis for the rate of profits is a combination of three pairs of wage curves intersecting over the axis for the rate of profits. These three fluke cases are the partitions between regions 1 and 2, regions 4 and 5, and regions 1 and 5. The partition between regions 2 and 3 is associated with the fluke case of three wage curves intersecting at a non-negative rate of profits. The partition between regions 3 and 4 illustrates a fluke switch point specific to models of rent.

 Figure 2: A Part of the Parameter Space

Regions 2, 3, and 4 illustrate the possible non-uniqueness and non-existence of a cost-minimizing technique. For concreteness, consider the point in region 4 with the wage curves and variation in rent per acre illustrated in Figure 3. For rates of profits up to the first switch point, Alpha is cost-minimizing. Epsilon is cost-minimizing between the switch points, and Delta is also cost-minimizing for high rates of profits in this range. Beyond the second switch point, no technique is cost-minimizing. Whether or not land is scarce depends on the distribution of income.

 Figure 3: Wage Curves and Rent for Region 4

How can one determine which techniques are cost-minimizing for a given rate of profits? Given the technique and the rate of profits, the costs of the capital goods, the rent on land, and wages can be summed for a unit level for each process. Iron, steel, and corn inputs incur the going rate of profits in this sum. The difference between the revenues and this sum is the extra profits obtained in operating a process. By definition, no process comprising the technique yields extra profits. The technique is cost-minimizing if extra profits cannot be obtained by operating any other process.

For the parameters illustrated in Figure 3, extra profits are obtained by operating process IV or V at Alpha prices for a rate of profits greater than that at the first switch point. Alpha is only cost minimizing at a lower rate of profits. Figure 4 depicts the extra profits available from each corn-producing process at Delta and Epsilon prices. The range of rates of profits in which each technique is cost-minimizing is indicated, and these ranges overlap. For rates of profits immediately greater than the rate of profits at the second switch point, prices of production indicate that Epsilon should be adopted when prices of production for Delta prevail and that Delta should be adopted when prices of production for Epsilon prevail. This circuit is a manifestation of the non-existence of a cost-minimizing technique.

 Figure 4: Extra Profits for the Delta and Epsilon Techniques

Qualitative properties of the analysis of the choice of technique do not vary within each numbered region. In region 1, Alpha is cost-minimizing for all feasible rates of profits. Land is not scarce, and obtains no rent. For a high enough rate of profits in region 2, Alpha and Delta are both non-uniquely cost-minimizing. The wage curve for Delta slopes up and rent per acre decreases with the rate of profits when Delta is operated. The switch point for Alpha and Delta is at a positive wage. For any rate of profits greater the rate of profits at the switch point, no cost-minimizing technique exists. In region 3, a switch point exists on the wage frontier between Alpha and Epsilon, at a rate of profits greater than the minimum rate of profits for Delta. A range of the rate of profits remains at which Alpha and Delta are both non-uniquely cost-minimizing. Above the rate of profits at this switch point, the wage frontier resembles the wage frontier in Figure 3 at rates of profits greater than the minimum rate of profits for Delta. In region 5, Delta is no longer cost-minimizing at any feasible rate of profits. Alpha is cost-minimizing at a low rate of profits, and Epsilon is uniquely cost-minimizing at any feasible rate of profits greater than the rate of profits at the single switch point.

Whether or not land obtains a rent can depend on the distribution of income. For a low-enough rate of profits in regions 2, 3, 4, and 5, the first three processes are operated. Iron, steel, and corn are each produced with one process, and land obtains no rent. For a higher rate of profits, the Delta or Epsilon technique can be cost-minimizing. Corn is produced by two processes, and scarce land obtains a rent. Even if the requirements for use can feasibly be satisfied with some land not farmed, the cost-minimizing technique may be such that two processes are operated side-by-side on land, with no land lying fallow.

## Saturday, November 26, 2022

### An Extensive Rent Example

 Figure 1: A Wage Curves and Rent for an Example of Extensive Rent

This post is a rewrite of this. It is the third in a series, with the first here and the second here.

The analysis of the choice of technique in models of extensive rent can be based on the construction of wage curves, even though the outer envelope does not represent the cost-minimizing technique. The orders of fertility and rentability are emphasized here. The order of fertility is defined for specified techniques, in which a single quality of land is used in each technique and that land pays no rent. At a given rate of profits, the qualities of land are ordered by wages, with the most fertile land paying the highest wage. The order of rentability specifies the sequence of different qualities of lands from high rent per acre to low rent per acre. Both orders may vary with the wage or the rate of profits. Table 1 presents coefficients of production for an example.

 Input Iron Industry Corn Industry I II III IV Labor 1 a0,2 91/250 67/100 Type 1 Land 0 49/100 0 0 Type 2 Land 0 0 59/100 0 Type 3 Land 0 0 0 9/20 Iron 9/20 a1,2 9/10000 67/1000 Corn 2 6/125 27/100 3/20

How much corn can be produced is constrained by the available quantities of each type of land. Endowments of land and requirements for use must be among the givens to analyze the choice of technique in this example. Suppose one hundred acres of each type of land are available, and net output is somewhere between 321 and 443 bushels of corn. Then all three types of land must be farmed with the parameters specified in Figure 1. One type will be only partially farmed. The iron-producing process must be operated in each of the three economically viable techniques. Table 2 describes which type of lands are fully cultivated and which type of land is left partially fallow in each of the Alpha, Beta, and Gamma techniques.

 Technique Land Type 1 Type 2 Type 3 Alpha Fully farmed Fully farmed Partially farmed Beta Partially farmed Fully farmed Fully farmed Gamma Fully farmed Partially farmed Fully farmed

For a given technique, the rent on a type of land only partially farmed is zero since it is not scarce. The wage curves in the left pane in Figure 1 are constructed, for each technique, from the price equations provided by the iron-producing process and the corn-producing process for the land that pays no rent. The choice of technique depends on both income distribution and requirements for use (Quadrio-Curzio 1980). Suppose the rate of profits is taken as given. Then the r–order of efficiency or fertility is the order of the wage curves downwards, until requirements for use are satisfied. Since all three types of land must be somewhat cultivated in the example, the wage frontier is the inner envelope of the wage curves. For the illustrated parameters, Alpha is cost-minimizing, and the order of fertility between the switch points is Type 1, Type 2, and Type 3 lands.

One can calculate the cost of capital goods at the given rate of profits and the cost of labor inputs for corn-producing processes on each of Type 1 and Type 2 lands. Since coefficients of production are specified per bushel corn produced, the revenues from each of these processes, at a unit level, are the same as the process operated on Type 1 land. Rent is the difference between revenues and non-land costs on Type 1 and Type 2 lands. Rent per acre is plotted in the right pane for Figure 1. The order of rentability is the order of lands by rent per acre. The order of rentability is the same as the order of fertility between switch points for the given parameters.

The two fluke switch points in Figure 1 do not lie on the (inner) wage frontier. The maximum rate of profits for the wage curve for Alpha is the maximum rate of profits for this example. One of the switch point for the wage curves for the Beta and Gamma techniques is at this maximum rate of profits. As seen in Figure 2, this is another edge case, a fluke that arises in models of extensive rent. By the way, at other parameter ranges the curves for rent per acre as a function of the rate of profits in the right pane in Figure 1 can intersect. The order of rentability can vary with distribution, and fluke cases in which these curves intersect at a maximum rate of profits or a rate of profits of zero can arise.

 Figure 2: The Parameter Space for an Example of Extensive Rent

To the southwest in Figure 2, one switch point between the wage curves for Beta and Gamma has vanished over the wage axis, and the other is at a rate of profits exceeding the maximum rate of profits for Alpha. The order of fertility matches the order of rentability. The northeast is of a reswitching example. For such small perturbations, the order of rentability does not change in this example. Thus, the order of fertility matches the order of rentability only at intermediate rates of profits with reswitching here. The northwest and southeast also illustrate parameters for which the order of fertility varies with the rate of profits. The order of fertility varies from the order of rentability at one or the other extreme, as indicated.

## Tuesday, November 22, 2022

### Fixed Capital And The Emergence Of Reswitching

 Figure 1: A Wage Frontier With A Fluke Switch Point

This post is a rewrite of this, without the attempt to draw a connection to structural economic dynamics. This is the second post in a series, starting with this.

A fluke example with fixed capital illustrates the emergence of the reswitching of techniques. Table 1 presents coefficients of production in a perturbation of an example from Schefold (1980). With the first process, workers, under the direction of mangers of firms, manufacture new machines. The remaining two processes are used to produce corn. The last process requires an input of an old machine, which is jointly produced with corn by the second process. Corn is both a consumption good and a capital good, insofar as it is an input into all three processes.

 Input Machine Industry Corn Industry One Process Another Process Labor 1/5 a0,2 7/5 Corn 1/8 a1,2 7/20 New Machines 0 1 0 Old Machines 0 0 1 Output Corn 0 1 1 New Machines 1 0 0 Old Machines 0 1 0

The choice of technique corresponds here to the choice of the economic life of the machine. This lifetime is truncated to one year for the Alpha technique, while the machine is operated for its full physical life of two years under the Beta technique. In a pure fixed capital model, the choice of technique can be analyzed by the construction of the wage frontier. The cost-minimizing technique at a given rate of profits has a wage curve on the outer frontier, as illustrated by Figure 1 for a specified parametrization. Managers of firms are willing to operate the machine for two years for any feasible rate of profits. At the maximum wage or a rate of profits of zero, the Alpha technique is also cost-minimizing. The single switch point is a fluke in two ways. First, it lies on the wage axis. Second, the wage curves are tangent at the switch point.

Figure 3 depicts a part of the parameter space for this example. A thin wedge between two partitions extends to the southeast of the point for the parameters corresponding to Figure 1. At the upper edge of this wedge, the two wage curves for the techniques are tangent at a switch point. The example is of reswitching below this partition and within this wedge. At the lower edge of this wedge, the switch point with the lower rate of profits is on the wage axis.

 Figure 2: The Parameter Space for an Example with Fixed Capital

Reswitching, in this example of fixed capital, is connected to the economic life of a machine. The economic life is the full two years here for a low and high rate of profits. Truncation occurs for a range of intermediate rates of profits. The specification of which technique is cost-minimizing can be consistent with vastly different functional distributions of income, with another technique being cost-minimizing for less extreme distributions

The switch point at the higher rate of profits in the reswitching region of the parameter space illustrates capital-reversing. Around this switch point, a lower rate of profits is associated with the adoption of a less capital-intensive cost-minimizing technique. At any rate of profits, inputs into production in a stationary state can be evaluated at prices of production, and these evaluations can be summed for each technique. The ratio of capital per worker, for example, is an index of the capital intensity of a technique. A more capital-intensive technique produces more output per worker, but its adoption is not necessarily encouraged by a lower rate of profits or interest rate (Harris 1973). In other words, a higher wage is associated with the adoption of a technique that requires a greater input of labor per bushel corn produced net throughout the economy. Capital-reversing has been shown to occur in other examples without reswitching on the wage frontier. Harcourt (1972) surveys the controversy in which economists, such as Paul Samuelson and Robert Solow, in Cambridge, Massachusetts, struggled to accept these conclusions drawn by other economists, such as Joan Robinson and Piero Sraffa, at the University of Cambridge.

Consider the region to the southwest in Figure 2. A single switch point exists on the wage frontier. Around this switch point, a lower rate of profits is associated with the adoption of a technique with a greater value of capital per person-year and a greater output per worker. Nevertheless, truncating the operation of the machine for one year is associated with a more capital-intensive technique. The demonstration of the invalidity of Austrian capital theory does not even need the phenomena of reswitching and capital-reversing.

## Thursday, November 17, 2022

### The Emergence Of The Reverse Substitution Of Labor

 Figure 1: A Wage Frontier With Two Fluke Switch Points

This post is a rewrite of this, without the attempt to draw a connection to structural economic dynamics.

This post presents an example with circulating capital alone. Table 1 presents the technology for an economy in which two commodities, iron and corn, are produced. Managers of firms know of one process for producing iron and two for producing corn. Each process is specified by coefficients of production, that is, the required physical inputs per unit output. The Alpha technique consists of the iron-producing process and the first corn-producing process. Similarly, the Beta technique consists of the iron-producing process and the second corn-producing process. At any time, managers of firms face a problem of the choice of technique.

 Input Iron Industry Corn Industry Alpha Beta Labor a0,1 = 1 a0,2α = 16/25 a0,2β Iron a1,1 = 9/20 a1,2α = 1/625 a1,2β Corn a2,1 = 2 a2,2α = 12/25 a2,2β = 27/400

Two parameters are not given numerical values in this specification of technology. The approach taken here is to examine a local perturbation of parameters in a two-dimensional slice of the higher dimensional parameter space defined by the coefficients of production in particular numeric examples. With wages paid out of the surplus product at the end of the period of production, the wage curves for the two techniques are depicted in Figure 1 for a particular parametrization of the coefficients of production. The Beta technique is cost-minimizing for any feasible distribution of income. If the wage is zero and the workers live on air, the Alpha technique is also cost-minimizing.

A switch point is defined in this model of circulating capital to be an intersection of the wage curves. These switch points, for the particular parameter values illustrated in Figure 1, are fluke cases. Almost any variation in the model parameters destroys their interesting properties. A switch point exists at a rate of profits of -100 percent only along a knife edge in the parameter space (Figure 2). Likewise, a switch point exists on the axis for the rate of profits only along another knife edge. The illustrated example, with two fluke switch points, arises at a single point in the parameter space, where these two partitions intersect.

 Figure 2: The Parameter Space for the Reverse Substitution of Labor

Figure 2 depicts a partition of the parameter space around the point with these two fluke switch points. Below the horizontal line, the switch point on the axis for the rate of profits has disappeared below the axis. The Beta technique is cost-minimizing for all feasible non-negative rates of profits. Above this locus, the Alpha technique is cost-minimizing for a low enough wage or a high enough feasible rate of profits.

In the northwest, the switch point at a negative rate of profits occurs at a rate of profits lower than 100 percent. Around the switch point at a positive rate of profits, a lower wage is associated with the adoption of the corn-producing process with a larger coefficient for labor. That is, at a higher wage, employment is lower per unit of gross output in the corn industry.

In the northeast of Figure 2, the switch point for a positive rate of profits exhibits the reverse substitution of labor. Around this switch point, a higher wage is associated with the adoption of a process producing the consumer good in which more labor is employed per unit of gross output. The other switch point exists for a rate of profits between -100 percent and zero. Steedman (2006) presents examples with this phenomenon in models with other structures.

Qualitative changes in the wage frontier exist in the parameter space away from the part graphed in Figure 2. The analysis presented here is of local perturbations of the depicted fluke case.

## Tuesday, November 15, 2022

### Scholarly Socialists During The Second International

Socialism became a mass movement in many European countries about the time of the heyday of the Second International. Many leaders of these movements and those struggling for leadership produced works of scholarship, albeit often with an activist spirit. I think of, for example:

• Eduard Bernstein. The Preconditions of Socialism and the Tasks for Social Democracy.
• Nikolai Bukharin. The Economic Theory of the Leisure Class.
• Richard B. Day and Daniel F. Guido (eds.). 2018. Responses to Marx's Capital. Brill
• Rudolph Hilferding. Böhm-Bawerk's Criticism of Marx.
• Rudolph Hilferding. Finance Capital.
• Karl Kautsky. The Social Revolution.
• Karl Kautsky. The Path to Power.
• Antonio Labriola. Essays on the Materialist Conception of History.
• Rosa Luxemburg. The Accumulation of Capital.
• G. V. Plekhanov. The Development of the Monist Theory of History.
• George Bernard Shaw (ed.). Fabian Essays in Socialism.
• Georges Sorel. Reflections on Violence.

I do not give copyright dates because I am unsure of the publication dates of some in the original german, italian, or russian. The Day and Guido work is a collections of essays from the time. This is hardly a comprehensive list of the literature of the period. All of these works, published before the October revolution, take Marx as serious and important. The authors were not academics, but I am not sure that there was a solid border between academia and politics at the time. Socialists, in works of scholarship, engaged with those developing the then new-fangled marginalist economic theory.

Collections
• Jukka Gronow. 2016. On the Formation of Marxism. Brill.
• M. C. Howard and J. E. King. 1989. A History of Marxian Economics, vol. 1. Princeton.
• Ian Steedman (ed.). 1995. Socialism and Marxism in Economics: 1870 - 1930. Routledge.

## Saturday, November 12, 2022

### Events Without Probability

1.0 Introduction

In a common model, probability theory assigns a number between zero and one to events. But some events cannot be assigned a probability, even zero. Nobody understands probability, in some sense.

This post is not about economics, although these ideas do have an application in mainstream economics. It is not at all novel. This is one of my favorite proofs in all of math, typically taught sometimes after an introductory mathematical analysis class. My undergraduate class in probability and statistics only alluded to measure theory.

2.0 An Introduction to Probability and a Overview

Suppose one has some sort of repeatable experiment, like rolling a pair of dice, dealing out a five-card poker hand, or spinning a spinner. The set of all possible outcomes is the sample space. Consider a subset of the sample space, such as all rolls in which the pair add up to seven, the hand is a straight flush, or the spinner stops at an angle between zero and 180 degrees. That subset is called an event. If all points in the sample space have the same probability of outcome, the probability is the ratio of the size of the subset for the event to the size of the sample space. (I suppose, more generally, this definition works for non-uniform distributions where the "size" includes weights, so to speak.) Probability is a set function, that is it maps each set in some sort of collection of subsets of the sample space into the real numbers.

This definition works well when the sample space contains a finite number of points. The "size" of a set is then just the number of points in the set. But consider the spinner example, where, with appropriate scaling of angles, the sample space is any real number in the interval [0, 1]. The number of points in a range of scaled angles [a, b] and in the sample space is, in both cases, uncountably infinite. Clearly, one needs some other concept of size here. And the concept of a probability measure provides the needed notion.

The demonstration of the existence of a non-measurable set validates the claim in the introduction. In the proof, the interval [0, 1] is partitioned into an uncountably infinite number of sets, each with only a countably infinite number of points in each set. The axiom of choice is used to "construct" from this partition another partition of the interval into a countably infinite number of sets, each with an uncountably infinite number of points. All of these sets have the same measure, and that measure must add up over the countably infinite number of sets to unity. But that measure can thus be neither zero nor non-szero. So some events exist, given the axiom of choice, without a probability.

3.0 Some Properties of a Measure

I use m(S) to denote the measure of a set. For this post, I consider measures with the following properties:

• The measure of an interval is merely the length of the interval:
m([a, b]) = b - a
• The measure of the empty set is zero:
m(∅) = 0
• The measure of any set S is non-negative. For all S
m(S) ≥ 0
• The measure of a set is translation invariant. Let S * x denote the set formed by adding x to each element S modulo one. (This definition keeps the translation of a set in the unit interval within the unit interval.) For all S and x:
m(S * x) = m(S)
• The measure of a set is countably additive. Let S1, S2, S3, ... be a sequence of disjoint sets. That is, for any ij, SiSj = ∅. Then:
m(S1S2S3 ...) = m(S1) + m(S2) + m(S3) + ...

I have above selected the properties of specific kinds of measures. But they are all that is needed for the proof of the existence of unmeasurable sets.

In the spinner example, some non-empty sets have a measure of zero. For example, the probability that the spinner will stop at any given real number in the interval is zero. So is the probability that the spinner will stop at any element in a countably infinite set.

4.0 A Partition of the Unit Interval into Uncountably Infinite Equivalence Classes

Define an equivalence relation as follows. Let two real numbers x1, x2 be equivalent if and only if x2 - x1 is a rational number. Partition the real numbers into equivalence classes by this equivalence relation. The rational numbers is one such equivalence class. The set of real numbers that differ from the square root of two by a rational number is another equivalence class. The set of real numbers differing from the square root of three by a rational number is a third equivalence class. The number π generates a third equivalence class. In fact, there are an uncountable infinite number of equivalence classes, and each such class can be put into a one-to-one mapping to the set of rational numbers.

Consider the collection of sets formed by the intersections of these equivalence classes with the unit interval. This is now a partition of the unit interval. I guess this is easy to understand, as compared to what comes next in this proof.

5.0 An Application of the Axiom of Choice

I now apply the axiom of choice. Let S be a set that contains one and only one point from each of the equivalence classes. Thus, S is a subset of the unit interval containing an uncountably infinite number of points. The difference between any two points in S is an irrational number.

For each rational number r in the unit interval, form the translation S * r. The set of rational numbers in the unit interval is countably infinite. That is, these rational numbers can be ordered in a sequence r1, r2, r3, ...

Corresponding to this sequence is a sequence of sets S*r1, S*r2, S*r3, ... Each one of these sets is a translation of the set S. They all contain an uncountably infinite number of points, and they are all subsets of the unit interval. The intersection of any two of these sets is the empty set. Furthermore every point in the unit interval is in exactly one of these sets.

6.0 Finishing the Proof

The union of these disjoint sets is the unit interval, and the measure of the unit interval is unity. Thus:

m(S*r1S*r2S*r3 ...) = m([0, 1]) = 1 - 0 = 1

m(S*r1S*r2S*r3 ...) = m(S*r1) + m(S*r2) + m(S*r3) + ...

By translation invariance:

m(S*r1S*r2S*r3 ...) = m(S) + m(S) + m(S) + ...

Therefore:

1 = m(S) + m(S) + m(S) + ...

Suppose the measure of S were zero. Then we would have proven that zero equals one, an obvious contradiction. But suppose the measure of S were positive. Then the right hand size of the above equality would be infinity, another contradiction. So no measure can be assigned to the set S. In the model of a spinner, the set S represents an event, and no probability can be assigned to this event.

7.0 Conclusion

Has anybody proved the existence of an unmeasurable set with a proof that is not a variation of the above? Can such existence be proven without relying on a proof by contradiction? How would you show that all proofs of such existence must use the axiom of choice? I believe it is also the case that the existence of an unmeasurable set implies the axiom of choice, although I have never seen this proven.

Probabilities cannot be assigned to some events. You will almost certainly never encounter such events in practical applications.

Appendix A. Some Mathematical Background

The above post presumes knowledge that the rationals are countable, that the reals constitute an uncountable infinity, and that an equivalence relation yields equivalence classes that partition a set into nonoverlapping subsets.

A.1 The Rationals are Countable

A countably infinite set can be ordered to be in one-to-one correspondence with the natural numbers, {1, 2, 3, ... }. (Sometimes I begin with zero.) Consider the ordering of positive fractions in Table A-1. The subscripts in parantheses show the order. The fractions are the ratio of the column index to the row index. Obviously, this sequence can be repeated forever. But some rational numbers are repeated. For example, the third fraction in the sequence is 1/2. But r(12) is 2/4. So when enumerating the positive rationals, throw out these repeats. Call the resulting sequence r1, r2, r3, and so on.

 1 2 3 4 5 ... 1 r(1) = 1/1 r(2) = 2/1 r(6) = 3/1 r(7) = 4/1 r(15) = 5/1 ... 2 r(3) = 1/2 r(5) = 2/2 r(8) = 3/2 r(14) = 4/2 .. 3 r(4) = 1/3 r(9) = 2/3 r(13) = 3/3 ... 4 r(10) = 1/4 r(12) = 2/4 ... 5 r(11) = 1/5 ...

So at least the positive rational numbers are countable. Then consider the sequence 0, r1, -r1, r2, -r2, ... Thus, the rational numbers are countable.

A.2 The Reals are Uncountable

The uncountability of the reals are demonstrated by Cantor's diagonizability argument. It is a proof by contradiction.

I think it easiest to think of the proof with real numbers in binary notation. And it is sufficient to show the real numbers between zero and one are uncountable. Accordingly, suppose somebody claims to have a sequence of all the real numbers between zero and one, as in Table A-2. One constructs a new real number as follows. Let the first digit to the right of the binary point be 0, if the corresponding digit in x1 is 1; 1, if the corresponding digit is 0. Let the second digit to the right of the binary point in this new number be 0, if the corresponding digit in x2 is 1; 1, if the corresponding digit is 0. And so on. Given some arbitary sequence, I have now constructed a new real number that cannot appear in the sequence. For it differs from every number in the sequence by at least one digit. So no such sequence can be constructed for the real numbers.

 x1 0.1001010... x2 0.0001000... x3 0.1010010... x4 0.0111000... ... ...

This mathematics demonstrates there are different sizes infinities. Nobody know whether or not there is another size infinity between the rationals and the reals. I am not even sure what this question would mean. But, if you accept that the above makes sense, you probably accept that there are an infinity of sizes of infinity bigger than the reals.

A.3 Equivalence Relations

A relation on a set S is a set of ordered pairs of elements in the set. Informally, the ordered pairs denote the subset of the Cartesian product S x S where the relation is true. Let a ˜ b denote an equivalence relation. That is:

• The relation is reflexive. For all a in S:
a ˜ a
• The relation is symmetric. For all a, b in S:
If a ˜ b, then b ˜ a
• The relation is transitive. For all a, b, c in S:
If a ˜ b and b ˜ c, then a ˜ c

In a sense, an equivalence relation is a generalization of equality. Every equivalence relation generates an equivalence classes, and vice versa. An equivalence class C is a subset of S such that for all a, b in C, a is equivalent to b. Equivalence classes partition a set. Every point in the set is in one equivalence class and only in one equivalence class. Any two equivalence classes are disjoint; their intersection is the empty set.