Is The Order Of Efficiency Unique At A Given Rate Of Profits In Models With Multiple Agricultural Commodities?

1.0 Introduction

This post puts forth two research questions that I think I will never get to. And it puts forth one I have started on.

2.0 Is the Order of Efficiency Well-Defined with Mutliple Agricultural Commodities?

I have worked through an example with absolute, extensive, and intensive rent. The orders of efficiency and rentability are uniquely defined in this example. The example has an industrially produced commodity and one commodity, 'corn' produced on land.

Suppose more than one agricultural commodity exists. For example corn and barley can both be produced on land. But assume that no joint production, other than the use of land, exists. Would the order of efficiency still be uniquely defined at a given rate of profits? Would it depend on the composition of corn and barley in net output, as well as the level of net output?

I see even in my example, which techniques are feasible depend on the amount of iron and corn in net output. But would multiple agricultural commodities create more issues for defining the order of efficiency?

3.0 Can Competive Marginal Products Rank Inputs in the Opposite Order of their Productivity?

Frank Hahn argues that marginal productivity, rightly understood, is entirely consistent with Sraffa's model. He concentrated on the case of single production. with a discrete technology, the value of the marginal product of labor, for example, is an interval formed out of the right-hand and left-hand derivatives of production functions.

Hahn considers intertemporal equilibrium paths in which endowments are not given. Rather, they are found as part of the solution for steady state paths. I would rather take them as limit points of non-steady state paths, perhaps with saddle-point stability. But setting these points aside, I cannot see that Hahn is otherwise wrong. Marginal productivity by itself is not a theory of income distribution.

But how does marginal productivity apply to a combination of intensive and extensive rent? In my example, I find a range of the rate of profits in which three types of land are ordered by efficiency in exactly the opposite order as rent per acre. This does not seem consistent with J. B. Clark's idea that, with marginal products, you get rewarded for your contributions (or rather the contributions of the factors you own) to production.

4.0 Which Isues Arising From General Joint Production Arise In Models Combining Extensive Rent and Fixed Capital?

I give an example. In such models, the wage frontier for the cost-minimizing technique is not the outer fronter for wages curves. So I guess, it can be upward-sloping. Can the cost-minimizing technique be non-unique away from switch points? Can I find Bidard's market economy converging to a cycle, with no cost-minimizing technique existing? D'Agata has examples in a model of inetensive rent.

Bidard And Klimovsky's Fake Switch Point

Figure 1: Wage Curves with A Fake Switch Point

1.0 Introduction

I find that, in developing an example on fixed capital and extensive rent, I need to review fake switch points. I am relying on Bidard's book, not on his paper with Klimovsky.

In this post, I just run through their example. They should have been more explicit about which techniques are feasible in their exposition.

2.0 Technology

Bidard and Klimvsky's example is one of pure joint production. The technology consists of the three processes shown in Table 1. Each column shows the inputs of person-years of labor, bushels of corn, and number of hogs needed as inputs for a unit level of operation. The outputs of bushels corn and number of hogs are also shown. Since each process has an output of more than one commodity, this technology is one of joint production. I assume constant returns to scale, and that each process takes a year to complete.

Table 1: Three Processes Comprise the Technology

Input	Processes
	Process 1	Process 2	Process 3
Labor	a0,1 = 1	a0,2 = 1	a0,3 = 1
Corn	a1,1 = 20	a1,2 = 20	a1,3 = 30
Hogs	a2,1 = 20	a2,2 = 20	a2,3 = 30
Outputs
Corn	b1,1 = 21	b1,2 = 23	b1,3 = 36
Hogs	b2,1 = 27	b2,5 = 20	b2,3 = 34

Three techniques (Table 2) can be constructed from these processes, when neither corn or hogs are a free good. Unlike in single production, a single technique cannot produce any composition of net output.

Table 2: Techniques of Production

Technique	Processes
Alpha	1, 2
Beta	1, 3
Gamma	2, 3

Another issue arises in the choice of technique that does not occur with joint production. Suppose the processes in Alpha are being operated. And suppose extra profits can be obtained with the the third process at Alpha prices. Which process in Alpha should be replaced? If the first process is replaced, the Gamma technique is adopted. If the second process is replaced, the Beta process is adopted. In single production, each process is associated with a specific industry, producing a specific commodity. In this example of joint production, considerations of feasibility are sufficient to answer this question.

3.0 Requirements for Use, Prices and the Choice of Technique

I consider two cases. The composition of net output is specified in different proportions in the two cases. Two techniques, out of three, are feasible in each case. But which technique is infeasible varies.

A price system is associated with each technique. Each process in the technique yields an equation. I assume that wages are paid out of the surplus at the end of the year. These equations for prices of production show the same rate of profits being made in both operated processes. I take net output as the numeraire. A third and final equation specifies that the price of the numeraire is unity.

The price equations can be solved, given the rate of profits. The wage, the price of corn, and the price of hogs are each a function of the rate of profits for each technique.

The question posed by the example is whether the outer frontier of the wage curves corresponds to the cost-minimizing technique. It does not. But, in the example when it does not, the wage curve on the outer frontier is not feasible.

3.1 Case 1

Suppose the requirements for use, that is, net output consists of 0.38 bushels corn and 0.62 hogs. Then Alpha is infeasible, and Beta and Gamma are feasible.

Two fluke cases are of interest here. If net output consists of 3/8 bushels corn and 5/8 hogs, all three techniques are feasible. Only the second processes are operated for the Alpha and Gamma techniques. The first and third processes are operated at positive levels for Beta. On the other had, a net output of 3/5 bushels corn and 2/5 hogs is an edge case where Alpha is infeasible, with Beta and Gamma feasible. Only the third process is operated for Beta and Gamma.

For the remainder of this section, I consider the specific intermediate case where Alpha is infeasible. Figure 2 graphs the wage curves. Only one switch point exists, at a rate of profits of 10 percent. Figure 3 shows extra profits for each profits at Beta prices. Beta is cost-minimizing for a non-negative rate of profits up to 10 percent. After that, extra profits can be obtained by operating the second process at Alpha prices.

Figure 2: Wage Curves with Alpha Infeasible

Figure 3: Case 1, Extra Profits at Beta Prices

Figure 4 shows extra profits for each process at Gamma prices. Gamma is cost-minimizing from the switch point at a rate of profits of 10 percent to the maximum rate of profits. For rates of profits smaller than in this rate, extra profits can be obtained by operating process 1.

Figure 4: Case 1, Extra Profits at Gamma Prices

In this case, the wage curve on the outer envelope for a rate of profits between 10 percent and the maximum rate of profits of 18 percent is not the wage curve for the cost-minimizing technique. But it is a wage curve, Gamma, for an infeasible techinque. If only wage curves for feasible techniques were considered, the construction of the outer frontier of wage curves would be a correct analysis of the choice of technique.

3.1 Case 2

Suppose now that net output consists of 0.13 bushels corn and 0.87 hogs. Alpha and Beta are feasible. Gamma is infeasible. (Here a fluke case is a net output of 1/8 bushels corn and 7/8 hogs. Alpha and Beta remain feasible, but only the first process is operated in each case at a positive level. Processes 2 and 3 are operated at a level of 0.)

Figure 1, at the top of this post, shows the wage curves. In this case, the wage curves on the outer frontier intersect at two points, the switch point at 10 percent and the fake switch point at approximately 13 percent.

The price system is identical among techniques at a switch point. Figure 5 shows how the price of corn varies with the rate of profits for the techniques in this case. Figure 6 shows how the price of hogs vary. You can see the switch point on these graphs of prices. No manifestation of the fake switch point is apparent.

Figure 5: Case 2, Price of Corn

Figure 6: Case 2, Price of Hogs

You can graph extra profits for each process, at prices for each technique, here too. Figure 7 graphs extra profits for Alpha. Alpha is cost-minimizing from the switch point at a rate of profits of 10 percent to the maximum at 20 percent. And Figure 8 demonstrates that Beta remains cost-minimizing for rates of profits below 10 percent.

Figure 7: Case 2, Extra Profits at Alpha Prices

Figure 6: Case 2, Extra Profits at Beta Prices

So Bidard and Klimovsky's claim that an intersection of wage curves on the outer frontier can be a fake switch point. In their example, one of the wage curves at the fake switch point is for an infeasible technique. And when the wage curve for the cost-minimizing technique lies below the outer frontier, the curve on the frontier is for that infeasible technique.

4.0 Conclusion

This example illustrates the interaction between requirements for use and the feasibility of techniques of production in models of joint production. The appearance of the wage curves depends on the numeraire. Bidard does not tie the numeraire to the requirements for use, as I do. He considers numeraires (one bushel corn for one case and one hog for the other) which cannot be feasibly produced with any technique. Likewise, Bidard does not consider feasibility. He discusses the angles between certain vectors in quantity space. I do not understand this without more studying, but perhaps it is a matter of feasibility.

Anyways, the example demonstrates the fallacy of analyzing the choice of technique by the construction of the outer wage frontier. In the first case, the cost-minimizing technique is not on the outer frontier for large rates of profits. The technique on the outer frontier is infeasible in this range.

The wage curve for the cost-minimizing technique is not always on the outer frontier in the second case either. Here, too the wage curve on the outer frontier is for a technique infeasible in this range. Furthermore, a fake switch point appears. The wage curves on the outer frontier intersect at a point where the cost-minimizing technique does not change and is unique. The prices of corn and hogs vary with the technique at the fake switch point. This variation is not possible in the single production models.

Which technique is cost-minimizing at a given rate of profits does not depend on the numeraire when the numeraire is disassociated with the net product.

I did not consider techniques in which only one process is operated to satisfy requirements for use. Perhaps there are such techniques in which one good is overproduced and becomes a free good.

References

• Bidard, Christian. 2004. Prices, Reproduction, Scarcity. Cambridge: Cambridge University Press.
• Bidard, Christian and Edith Klimovsky. 2004. Switches and fake switches in methods of production. Cambridge Journal of Economics 28 (1): 88-97.

CCC: tl; dr

1.0 Introduction

You can easily find peopole on the Internet asking for a summary of the Cambridge Capital Controversy. An answer is easy.

Supply and demand is balderdash. If you want to understand markets under capitalism, you might as well throw away most microeconomic textbooks and most introductory textbooks.

2.0 Some Expansions

No consistent, valid model in which more than one commodity is produced supports the following two mistaken stories.

Suppose the tastes of those making decisions for households change. They become more future-oriented, more willing to defer consumption. The supply of capital has increased. With an increased supply, the interest rate is driven down. The firms ultimately choose to adopt more capital-intensive techniques. They tend to equip workers with more machinery and to run exisiting machinery longer.

Suppose, again, that the tastes of those making decisions for households change. Workers now prefer consumption over leisure more than they did before. The supply of labor has increased. The real wage is driven down. Firms ultimately choose to adopt more labor-intensive techniques. Equilibrium employment in competitive markets thereby increases.

Numerical examples of capital-reversing and other so-called capital-theoretical paradoxes or perversities are enough to demonstrate that the above stories do not follow from the assumptions of mainstream economics.

3.0 Extension to Technical Discussions

Well-educated economists know that their theory does not support the causal stories in the textbooks. I concentrated above on factor markets. I now go into more technical points. I think that if my powers of exposition and understanding were much greater, this section would still not be clear.

The above refuted stories can be augmented with stories about natural resources and about produced commodities. The assumptions of mainstream economics do not justify explaining equilibrium by the intersection of well-behaved supply and demand curves.

Demonstrations of capital-reversing, for example, are usually presented in open models of competitive, cost-minimizing firms. These models can be closed by assuming households are utility-maximizing. These closures include intertemporal utility-mximizing. Overlapping generations (OLG) models are examples.

Are long-run equilbrium models like this 'neoclassical'? Endowments of capital are not among the givens. The mixture and level of capital goods are found from solving the model. Likewise, the numeraire value of the capital stock is endogeneous, not a parameter.

The claim that, in equilibrium, the rate of interest is equal to the marginal product of capital might be justified as applying to Champernowne's chain index measure of the quantity of capital. This chain index, basically, excludes price Wicksell effects from the measure of the quantity of capital. Both the interest rate and the value of this chain index are found from the solution of the model. They are not part of what needs to be known to find the solution. Furthermore, the chain index is not what is measured in empirical applications of the Solow-Swan model and in measurements of total factor productivity (TFP).

Another argument turns around how capital-theory paradoxes apply to models of intertemporal equilibria, if at all. Can a continuum of equilibria be found in either long-run models or models of intertemporal equilibrium? Do capital-theoretic paradoxes add anything to examination of the stability of intertemporal equilibrium, either for tattonement or for individual paths? Typically, stability cannot be demonstrated, and multiple equilibria exist. No reason exists to expect paths to approach steady states. J. Barkley Rosser Jr. argued that the reswitching of techniques is manifested in intertemporal equilibrium as a cusp catastrophe. Personnally, I think Michael Mandler is more correct than Pierangelo Garegnani about stability.

I finally bring up that you do not have to close long run models with intertemporal utility-maximization. Richard Kahn, Nicholas Kaldor, Luigi Pasinetti, and Joan Robinson had closures related to the Cambridge equation. Or you can take the wage as a matter of social conventions or norms. Or, perhaps, you can have a monetary theory of distribution, in which the monetary authority sets the interest rate. Stephen Marglin had a overdetermined closure that explained stagflation. Questions exist about how some of these closures relate to a generalization of John Maynard Keynes' principle of effective demand to the long run.

I see I have left out debates over the the history of more than two centuries of political economy.

4.0 Conclusion

But the mainstream defenders in these discussion are not defending what is in the textbooks. The simple-minded depiction of well-behaved supply and demand curves determining equilibrium lacks any theoretical or empirical foundation.

Arguments And Debates Among Early Socialists

1. Introduction

Early socialists did not always agree. I provide an account of an idiosyncratic selection of a couple of Ricardian socialists, including a later dispute about priority and a debate with philosophical radicals, that is, liberals. I have been reading John Cassidy's new book. He has a chapter on William Thompson.

2. William Thompson, Karl Marx, Friedrich Engels, and Anton Menger

I start with William Thompson, the author in 1824 of An Inquiry into the Principles of the Distribution of Wealth Most Conducive to Human Happiness. Thompson built upon the ideas of Jeremy Bentham, William Godwin, and David Ricardo. Apparently, Thompson was to first to argue for equality of distribution on the basis of utilitarianism. An extra $100 has a lot more utility to a poor man than a rich one. He coined the term "surplus value", and his political economy had a structure much like Karl Marx's later system, with some differences. Thompson advocated, along with Robert Owen, for mutual co-operation, as in intentional communities.

"it was Ricardo's crude generalisations which gave modern socialism its fancied scientific basis, and provoked, if they did not justify, its revolutionary form… Ricardo ... did more than any intentionally socialist writer to sap the foundations of that form of society which he was trying to explain, and which he believed to be the typical and natural, if not, indeed, the ideal social state...

Thompson was only one of a series of socialist writers, culminating in Marx and Lassalle, who take the Ricardian position as the very basis of their argument. His first section has the familiar Ricardian ring. 'Wealth is produced by labor: no other ingredient but labor makes any object of desire an object of wealth. Labor is the sole universal measure, as well as the characteristic distinction of wealth.' Give the word 'labour' its popular meaning, and it is merely an affair of logic to deduce a large part of modern socialism from this position. Whatever qualifications Ricardo may have made upon it in his own mind, ninety-nine readers out of a hundred took him literally, and the main impression left by his book was that while wealth was almost exclusively due to labour, it was mainly absorbed by rent and other payments to the unproductive classes. This was the text which Thompson and the English socialists proceeded to elaborate." -- H. S. Foxwell

Anton Menger says that Marx basically copied Thompson and other Ricardian socialists:

"Marx is completely under the influence of the earlier English socialists, and more particularly of William Thompson. Leaving out of account the mathematical formulae by which Marx rather obscures than elucidates his argument, the whole theory of surplus value, its conception, its name, and the estimates of its amount are borrowed in all essentials from Thompson's writings. Only Marx, in accordance with the aim of his work, pays special attention to the one form of unearned income (interest on capital), and fails to give either that juridical criticism of private property in instruments of production and useful commodities which is the necessary supplement of the theory of surplus value, or a rigorous exposition of the right to the whole produce of labour. In all these respects Marx is far inferior to Thompson." -- Anton Menger

Engels would have none of it:

"In order, to drag Marx down, his achievements are attributed to other socialists in whom no one is interested, who have vanished from the scene and who have no political or scientific importance any longer. In this way they hope to dispose of the founder of the proletarian world view, and indeed the world view itself. Mr. Menger undertook the task. People are not professors for nothing. They want to make their mark, too...

The present social order gives landowners and capitalists a 'right' to part - the bulk - of the product produced by the worker… So whoever first said that the present right of those who own the soil and the other means of production to part of the proceeds of labour is a wrong is the great man, the founder of 'scientific' socialism. And these men were Godwin, Hall and Thompson. Leaving out all the interminable economic fripperies and getting to the legal residue, Menger finds nothing but the same assertion in Marx. Consequently Marx simply copied these old Englishmen, particularly Thompson, and took care to keep quiet about his source. The proof has been adduced.

We give up any attempt to make this hidebound lawyer understand that nowhere does Marx demand the 'right to the full proceeds of labour', that he makes no legal demands of any kind at all in his theoretical works. Even our lawyer seems to have a faint inkling of this when he reproaches Marx for nowhere giving 'a thorough presentation of the right to the full proceeds of labour'.

In Marx's theoretical studies legal right, which always merely reflects the economic conditions prevalent in a specific society, is only considered as a matter of purely secondary importance; his main concern is the historical justification for certain conditions, modes of appropriation and social classes in specific ages, the investigation of which is of prime importance to anyone who sees in history a coherent, though often disrupted, course of development rather than, as the eighteenth century did, a mere muddle of folly and brutality. Marx views the historical inevitability of, and hence the justification for, the slave-owners of classical times, the feudal lords of the Middle Ages, etc., as the lever of human development for a limited historical period. He thereby also recognises the temporary historical justification for exploitation, for the appropriation of the product of labour by others. Yet at the same time he demonstrates that not only has this historical justification disappeared, but that the continued existence of exploitation in any form, far from furthering social development, is daily impeding it more and more and involving it in increasingly violent collisions. Menger's attempt to force these epoch-making historical investigations into his narrow, legalistic Procrustean bed only goes to show his total inability to understand things that go beyond the narrowest legal horizon." – Friedrich Engels and Karl Kautsky (1887).

Both Menger and Engels and Kautsky have a lot more to say.

3. William Thompson and J. S. Mill

William Thompson and others had a public debate with John Stuart Mill in 1825. Mill was then a teenager, a liberal, a philosophical radical, and an utilitarian.

"There was for some time in existence a society of Owenites. called the Cooperative Society, which met for weekly public discussions in Chancery Lane. In the early part of 1825, accident brought Roebuck in contact with several of its members, and led to his attending one or two of the meetings and taking part in the debate in opposition to Owenism.

Some one of us started the notion of going there in a body and having a general battle: and Charles Austin and some of his friends who did not usually take part in our joint exercises, entered into the project. It was carried out by concert with the principal members of the Society, themselves nothing loth, as they naturally preferred a controversy with opponents to a tame discussion among their own body. The question of population was proposed as the subject of debate: Charles Austin led the case on our side with a brilliant speech. and the fight was kept up by adjournment through five or six weekly meetings before crowded auditories, including along with the members of the Society and their friends, many hearers and some speakers from the Inns of Court.

When this debate was ended, another was commenced on the general merits of Owen's system: and the contest altogether lasted about three months. It was a lutte corps-@agrave;-corps between Owenites and political economists, whom the Owenites regarded as their most inveterate opponents: but it was a perfectly friendly dispute. We who represented political economy had the same objects in view as they had, and took pains to shew it; and the principal champion on their side was a very estimable man, with whom I was well acquainted, Mr. William Thompson. Of Cork, author of a book on the Distribution of Wealth, and of an Appeal in behalf of women against the passage relating to them in my father's Essay on Government.

Ellis, Roebuck, and I, took an active part in the debate, and among those from the Inns of Court who joined in it I remember Charles Villers. The other side obtained also, on the population question, very efficient support from without. The well known Gale Jones, then an elderly man. made one of his florid speeches; but the speaker with whom I was most struck, though I dissented from nearly every word he said, was Thirlwall, the historian, since Bishop of St. David's, then a Chancery barrister, unknown except by a high reputation for eloquence acquired at the Cambridge Union before the era of Austin and Macaulay. His speech was in answer to one of mine. Before he had uttered ten sentences. I set him down as the best speaker I had ever heard, and I have never since heard any one whom I placed above him." – J. S. Mill, Autobiography

In his autobiography, Mill makes clear that those on his side are his friends from university.

3. William Thompson and Thomas Hodgskin

Thompson had a disagreement with Thomas Hodgskin who published in 1825, Labour Defended Against the Claims of Capital. Hodgskin's explanation for profits is a matter of selling commodities over their value, with force and monopoly power.

"But if [Hodgskin] is in favour of competition as the principle by which to determine the division of labour's share between the various ranks of labourers, he is for combination against capital in order to make labour's share as large as possible. By combining, the journeymen 'may reduce or destroy altogether the profit of the idle capitalist ... but they will augment the wages and rewards of industry, and will give to genius and skill their due share of the national produce.'" – H. S. Foxwell This is the sort of explanation and remedy that Thompson rejected. His 1827 book Labour Rewarded is an answer.

"Thompson urges that 'individual competition is incompatible with equal remuneration, as it is also with securing to labor the entire products of its exertions'. 'The author of Labour Defended stands alone, as far as. I know, amongst the advocates of Individual competition, in even wishing that labor should possess the whole of the products of its exertions. All other advocates of individual competition look on the notion as visionary, under the Competitive System'. We know Thompson's solution of the difficulty. Labourers must become capitalists, and unite in communities to regulate their own labour. To ascertain for each the exact product of his own labour is impracticable. If this could be done, then justice would give each individual a property in that product. But moral considerations would force him to share that product with others. The human race could not otherwise be preserved. This voluntary distribution is best carried out under the equitable arrangements of co-operative communities, with their regulated exchanges. 'It is on the regulation of exchanges', he concludes, 'that the industrious classes must depend for realising the general proposition that "the whole produce of labour should belong to the labourer."'" – H. S. Foxwell

We see here the distinction between each individual worker obtaining his product and the working class as a whole obtaining and deciding on the use of their product. Thompson advocates the latter.

4. William Thompson and Robert Owen

Thompson had a later disagreement with Robert Owen. This disagreement, at the 1832 Co-operative Congress in London, arose after Owen's experiences in the USA, at New Harmony, Indiana. Owen thought co-operatives needed much larger initial investments than Thompson.

Selected References

• John Cassidy. 2025. Capitalism and Its Critics: A History from the Industrial Revolution to AI. Farrar, Straus, and Giroux.
• Friedrich Engels and Karl Kautsky. 1887. Lawyer's Socialism. Die Neue Zeit.
• Anton Menger. 1899. The Right to the Whole Produce of Labour. With an introduction by H. S. Foxwell.

Fixed Capital And Extensive Rent

This post is a problem statement.

Models of the production of commodities with circulating capital alone have certain nice properties. I refer to models in which commodities are produced by means of commodities, with a certain circular structure in production. Direct labor inputs are assumed to be necessary to operate each process in the technology.

The choice of technique can be analyzed in models with circulating capital alone by constructing the outer envelope of the wage curves for each technique. Each wage curve slopes down. The wage, for a technique, is lower the higher the rate of profits. The cost-minimizing technique at a given rate of profits is unique, except at switch points. The "determination of the cost-minimising technique is independent of the structure of requirements for use" (Huang 2019). The wage and prices of production are unique functions of the rate of profits. If a technique exists with a defined wage and prices of production at a given rate of profits, then a cost-minimizing technique exists. A market algorithm (Bidard 1990) converges, without going into a cycle.

None of these properties necessarily hold in models with joint production. For example, Bidard & Klimovsky (2004) define fake switch points as intersections on the outer wage frontier at which the cost-minimizing technique does not vary.

But they do in a model of pure fixed capital, with the exception that wage curves can slope up when not on the frontier. These models are non-interlocked systems in which old machines cannot be transferred among sectors. Each process produces exactly one finished good, such as a consumption good; a good used as circulating capital; or a new, possibly long-lived machine. Old machines are intermediate goods. Old machines cannot be consumer goods. Every finished good has exactly one primary process for producing it, in which an intermediate good does not enter as an input. In each sector, the secondary processes completely use up the old machine produced by the primary process in that sector, with no other intermediate goods as inputs. They produce the same finished good, possibly jointly with an intermediate good. Joint utilization of machines does not exist in any process. Old machines may be freely disposed of; no cost arises in junking a machine, including before its technical life is complete.

Most of the properties of circulating capital also hold in models of extensive rent. Extensive rent occurs when multiple types of land must be cultivated to satisfy requirements for use. Only one process is operated on each type of land. All but one of the types of cultivated land are fully farmed, except in fluke cases, to the extent of their endowment. With only one price prevailing for corn and only one rate of profits being obtained in the system of prices of production, different amounts of rent per acre must be paid on the different types of land that are fully farmed. The type of land that is only partially farmed is not scarce and does not pay a rent. The cost-minimizing technique can be found from wage curves, but it does not correspond to the technique on the outer frontier. Requirements for use determine which techniques are feasible, and only a feasible technique can be cost-minimizing.

My claim is that a model combining fixed capital and extensive rent lacks more properties characteristic of circulating capital than either does alone. For example, the wage frontier, defined by the wage curves for cost-minizing techniques at each rate of profits, can slope upward. A numerical example can demonstrate this. So I am curious if an elaboration of the model specified in this post works for this purpose.

Tables 1 and 2 specify the technology for a simple model producing multiple commodities and combining fixed capital and extensive rent. A numerical example results by setting each coefficient of production not equal to zero or unity to some positive value.

Table 1: Inputs for Processes Comprising the Technology

Input	Industry
	Machine	Corn
	I	II	III	IV	V
Labor	a0,1	a0,2	a0,3	a0,4	a0,5
Type 1 Land	c1,1 = 0	c1,2	c1,3	c1,4 = 0	c1,5 = 0
Type 2 Land	c2,1 = 0	c2,2 = 0	c2,3 = 0	c2,4	c2,5
Corn	a1,1	a1,2	a1,3	a1,4	a1,5
New Machines	a2,1 = 0	a2,2 = 1	a2,3 = 0	a2,4 = 1	a2,5 = 0
Type 1 Old Machines	a3,1 = 0	a3,2 = 0	a3,3 = 1	a3,4 = 0	a3,5 = 0
Type 2 Old Machines	a4,1 = 0	a4,2 = 0	a4,3 = 0	a4,4 = 0	a4,5 = 1

Table 2: Outputs for Processes Comprising the Technology

Output	Industry
	Machine	Corn
	I	II	III	IV	V
Corn	b1,1 = 0	b1,2	b1,3	b1,4	b1,5
New Machines	b2,1 = 1	b2,2 = 0	b2,3 = 0	b2,4 = 0	b2,5 = 0
Type 1 Old Machines	b3,1 = 0	b3,2 = 1	b3,3 = 0	b3,4 = 0	b3,5 = 0
Type 2 Old Machines	b4,1 = 0	b4,2 = 0	b4,3 = 0	b4,4 = 1	b4,5 = 0

In the tables, each column specfies the inputs and outputs of corn, new machines, and old machines of each type needed to operate that process at a unit level. Inputs of labor and each of the two types of land are also specified. I assume constant returns to scale, up to the limits imposed by endowments of land. Each process requires the same time, a year, to complete. The first process uses inputs of labor and corn to manufacture new machines. The second and third processes produce corn on the first type of land. The remaining two processes also produce corn, but on the other type of land.

Corn is the numeraire and the only consumption good. The total acres of each type of land are also part of the data. Requirements for use are specified by the quantity of corn in the net output of the economy.

Table 3 lists the techniques of production available. In Alpha, Beta, Gamma, and Delta, land is not scarce. Only one type of land is farmed, and the quantity farmed does not exceed its endowment. Beta differs from Alpha in that a machine of the first type is used for its full physical life. Likewise, Delta differs from Gamma in the same way for a machine of the second type. Both types of land are farmed in the remaining eight techniques, and ownership of one type of land obtains a rent. The techniques vary in which land receives a rent and in the economic lifetime of a machine.

Table 3: Techniques of Production

Technique	Processes	Land
		Type 1	Type 2
Alpha	I, II	Partially farmed	Fallow
Beta	I, II, III	Partially farmed	Fallow
Gamma	I, IV	Fallow	Partially farmed
Delta	I, IV, V	Fallow	Partially farmed
Epsilon	I, II, IV	Fully farmed	Partially farmed
Zeta	I, II, III, IV	Fully farmed	Partially farmed
Eta	I, II, IV, V	Fully farmed	Partially farmed
Theta	I, II, III, IV, V	Fully farmed	Partially farmed
Iota	I, II, IV	Parially farmed	Fully farmed
Kappa	I, II, III, IV	Parially farmed	Fully farmed
Lambda	I,II, IV, V	Parially farmed	Fully farmed
Mu	I, II, III, IV, V	Parially farmed	Fully farmed

Which techniques are feasible for a given level of net output? What are the quantity flows? What are the price systems? Which techniques are cost-minimizing? Can I specify numerical values that illustrate interesting phenomena? These questions might be worth answering.

This combination of fixed capital and extensive rent should also demonstrate that approaches to the analysis of the choice of technique customized for each apply to their combination. If the price of some machines or rent on a farmed type of land is negative at a given rate of profits, that technique cannot be cost-minimizing at that rate, for example.

Three Cases In Which The Labor Theory of Value Is A Simple Theory Of Price

1.0 Introduction

This post should be mostly uncontroversial. The labor theory of value (LTV) explains prices in certain special cases.

2.0 An Economy of Self-Employed Artisans

First, consider a pre-capitalist economy with no employer and employees. Commodities are manufactured for the market by self-employed artisans. They might have titles like carpenters, weavers, or blacksmiths. In this case, market prices tend to be (proportional to) labor values.

Adam Smith described this case. Karl Marx can be read as presenting this case in chapter 1 of volume 1 of Capital. In that chapter, he has independent producers. He introduces concepts in a certain order. Can you find anything in that chapter about workers being paid wages?

3.0 When the Rate of Profits is Zero

I now turn my attention to a sort of capitalist economy. Employers hire workers for wages. The workers produce commodities for the employers to obtain the revenues from selling them on the markets.

But here the capitalists do not have the ability to generally obtain a profits. Market prices tend towards labor values.

Joseph Schumpeter wrote about this case in The Theory of Economic Development. He has profits being temporary, as innovating entrepreneurs destroy the smooth reproduction of a capitalist economy, until it tends to settle down again.

4.0 When Capital-Intensity Does Not Vary Among Industries

I now consider a capitalist economy in which the capitalists are generally able to obtain a rate of profits. Suppose capital-intensity does not vary among industries. The ratio of the total value of the capital goods used up in production to the labor directly used in producing each commodity is the same, whatever commodity is being produced. You might as well use labor values when adding up capital goods.

Then market prices tend towards labor values.

Karl Marx wrote about this case in much of the first volume of Capital. Obviously, he does not write about eigenvalues and Leontief input-output matrices.

5.0 Conclusion

Modern reformulations of classical political economy, of course, are not confined to the above special cases. I find of particular interest the mathematical dual of the last case. David Ricardo and Karl Marx discuss this dual case - not with that jargon. Ricardo's last essay, unpublished in his lifetime, was about that. Marx discussed this dual case towards the start of the third volume of Capital.

But really their interest was not in a theory of prices. Ricardo and Marx both investigated labor values in their explanations of profits, of the ability of capitalists to obtain returns on their investments.

Prohibitions On Tactics In Negotiating Your Wage

Pro-capitalists often depict wages as being determined by free negotiations between employers and employees. They ignore conventions and norms surrounding such negotiations. And laws regulate and prohibit various tactics, especially some that require collective action in solidarity with other workers.

In explaining these restrictions, I concentrate on vignettes in the United States of America. A theory of dual labor markets applies here. In the formal sector, you can expect benefits, paid vacations, defined limitations on working hours, and weekends off. In the other sector, not so much. I have seen talk about the precarity. It is clear that we need more labor unions in the USA. The United Auto Workers has been promising lately.

The rise of labor unions in the USA occurred against massive reactionary violence on the part of the minions of oligarchs, including the state. For example, President McKinley sent the Army to Idaho in 1899 to round up striking workers and others - they did not care who - and put them in the 'bullpen'. This was basically a concentration camp. The robber barons could also call on their own private police force. I refer especially to the Pinkerton Detective Agency.

I skip ahead to the National Labor Relations Act of 1935, also known as the Wagner act. This law legalized labor unions, collective bargaining, and strikes. It prohibited company unions. And it established the National Labor Relations Board (NLRB).

The Taft-Hartley Act was passed in 1947, over Truman's veto. It permitted states to pass so-called 'Right to Work' laws. It prohibited sympathy strikes and general strikes. Union leaders would like to sign contracts in which all jobs in certain grades or categories are union jobs. Employees hired for those jobs can be non-union, but they must pay union fees. A non-union worker might as well join the union to have their voice heard. The misnamed 'right to work' laws prohibit such free contracts. A sympathy strike is called by a union in solidarity with other workers. For example, pilots might go on strike when flight attendants are striking. You could imagine the workers in a city all going on strike in support of a political issue. But this is prohibited in the USA.

The above presents an overview of some elements of history in the USA in the setting in which workers 'freely' negotiate their wage. I focus on the prospect of labor organizing. A more comprehensive history would include the perception of those running the Federal Reserve that they should raise interest rates when wages rise and the general hostility to labor, particularly in the Republican party, since Reagan was president. Strangely enough, Reagan was the only president of a labor union (the Screen Actors Guild) ever to be elected president.

Ludwig Von Mises, Crackpot Conspiracy Theorist?

I get this from Sam Tanenhaus' new biography of William F. Buckley, Jr. The index entry for Von Mises is in error.

The John Birch Society was an extreme reactionary organization in the USA. Bob Welch was its leader and founded it near the start of 1959. Welch had a book, The Politician, later known as the Blue Book, that explained what he was about. According to Welch, George Marshall, Dwight Eisenhower, Earl Warren, the Dulles brother, and on and on were all conscious, dedicated Soviet agents.

Buckley had a magazine, National Review. And he was known for trying to show that American conservatives had views worth taking seriously. They are not all crackpots. So he tried to convey an impression that Bob Welch was not welcome in his movement.

The John Birch Society had a monthly magazine, American Opinion. Von Mises was on its board and had articles published in it. I have read quite a bit of Von Mises and about him too, mostly hagiographies. I never met this historical bit before.

Skilled Labor, Labor Values, Prices Of Production

1.0 Introduction

Suppose not all labor in a capitalist economy is 'unskilled'. Some jobs require specific skills that somehow command a premium. How can heterogeneous labor be handled in modern treatments of classical and Marxian political economy?

My inclination is just to assume that relative wages for different labor categories are stable. I read Ricardo and others as doing the same:

"I must not be supposed to be inattentive to the different qualities of labour, and the difficulty of comparing an hour's or a day's labour, in one employment, with the same duration of labour in another. The estimation4 in which different qualities of labour are held, comes soon to be adjusted in the market with sufficient precision for all practical purposes, and depends much on the comparative skill of the labourer, and intensity of the labour performed. The scale, when once formed, is liable to little variation." -- David Ricardo, Principles

But in this post, I want to consider a different approach. I assume that skilled labor is produced by prior training. This post is only a start.

2.0 Technology

I consider the simplest model of the production of commodities for making my point. Table 1 specifies the technology for this example. Each column shows the bushels corn, person-years of skilled labor, and the person-years of unskilled labor paid by the capitalist operating that process to produce one unit of output. I assume constant returns to scale.

Table 1: Technology

Input	Sector
	Training	Corn
Corn	a1,1 Bushels	a1,2 Bushels
Skilled Labor	c1,1 Person-Years	c1,2 Person-Years
Unkilled Labor	a0,1 Person-Years	a0,2 Person-Years
OUTPUT	One Person-Year	One Bushel

I assume the skilled labor produced by the training sector only has skills for the next year. After working for one year, a skilled worker needs to be retrained. In this way, the model resembles a model with only circulating capital and no fixed capital. The inputs to the training sector do not include the workers who emerge as skilled workers. Rather, those workers are customers, paying the capitalist, who obtains returns from operating the process in the training seector.

All coefficients of production are assumed to be positive. I assume the Hawkins-Simon conditions:

a_{1, 2} < 1

c_{1, 1} < 1

a_{1, 1} c_{1, 2} < (1 - a_{1, 2})(1 - c_{1, 1})

These asumptions ensure that the economy can produce a surplus product.

3.0 Labor Values

I first want to calculate labor values. Labor values are defined in terms of unskilled labor. Introduce the following two variables:

• v₁ is the (unskilled) labor value of corn, in unskilled person-years per bushel.
• v₂ is the (unskilled) labor value of skilled labor, in unskilled person-years per skilled person-year.

The labor value of a bushel corn is the sum of the labor values of the inputs needed to produce it.

v₁ = a_{0, 2} + a_{1, 2} v₁ + c_{1, 2} v₂

The specification of a process for training skilled labor allows for a parallel definition of the labor value of skilled labor:

v₂ = a_{0, 1} + a_{1, 1} v₁ + c_{1, 1} v₂

The above is a system of two equations in two unknowns. The following abbreviation is useful in setting out the solution:

denom = (1 - a_{1, 2})(1 - c_{1, 1}) - a_{1, 1} c_{1, 2}

The solution is:

v₁ = [a_{0, 1} c_{1, 2} + a_{0, 2}(1 - c_{1, 1})]/denom

v₂ = [a_{0, 1}(1 - a_{1, 2}) + a_{0, 2} a_{1, 1}]/denom

The Hawkins-Simon conditiones guarantee that labor values are positive. Despite the existence of heterogeneous labor, the labor value of corn, in terms of one type of labor is well-defined.

4.0 Prices of Production

I skip exploring various ratios and other aspects of the system of labor values. I introduce the following variables for prices of production:

• p₁ is price of corn, in numeraire-uints per bushel.
• w₁ is the wage of unskilled labor, in numeraire units per person-year.
• w₂ is the wage of skilled labor, in numeraire units per person-year.
• r is the rate of profits, a pure number.

The system of equations for prices of production includes an equation for corn:

(p₁ a_1,2)(1 + r) + c_1,2 w₂ + a_0,2 w₁ = p₁

In this equation, wages of skilled and unskilled labor are paid out of the surplus, at the end of the year. The following equation applies to the training sector:

(p₁ a_1,1)(1 + r) + c_1,1 w₂ + a_0,1 w₁ = (w₂ - w₁)/(1 + r)

The Right Hand Side is the present value of the skill premium obtained by a trained worker.

Specifying the numeraire removes one degree of freedom for the system of prices of production. One degree of freedom remains.If the wage of unskilled labor is taken as given, the system is closed.

5.0 Conclusion

Obviously, this analysis can go in many directions. What would it mean for the organic composition of of capital not to vary between the training sector and the corn-producing sector? How does Marx's account of exploitation work here? Can skilled workers exploit unskilled worker, as in Ian Steedman's book? How can I draw on the theory of fixed capital to account for skills lasting for multiple periods? Can I introduce risk and distinguish between the rate of profits and the interest rate used for time-discounting? How does the the theory of rent apply to inate talents? (Bootlickers will insist this is the dominant case.)

If you want to apply this approach empirically, you must answer some of these questions. The approach can be extended to more sectors and more types of labor. I have seen some input-output tables broken down to have two types of labor.

John Stuart Mill, Avowed Socialist

I have been reading some secondary literature on John Stuart Mill. He was explicitly against meritocracy, although that term was not available until Young's satirical novel. Here is Mill:

"If some Nero or Domitian were to require a hundred persons to run a race for their lives, on condition that the fifty or twenty who came in hindmost should be put to death, it would not be any diminution of the injustice that the strongest or nimblest would, except through some untoward accident, be certain to escape. The misery and the crime would be that any were put to death at all. So in the economy of society; if there be any who suffer physical privation or moral degradation, whose bodily necessities are either not satisfied or satisfied in a manner which only brutish creatures can be content with, this, though not necessarily the crime of society, is pro tanto a failure of the social arrangements. And to assert as a mitigation of the evil that those who thus suffer are the weaker members of the community, morally or physically, is to add insult to misfortune. Is weakness a justification of suffering? Is it not, on the contrary, an irresistible claim upon every human being for protection against suffering?" – J. S. Mill

The above is not necessarily an argument for socialism. It is consistent with Mill's investigation of what a society organized around private property might be.

In his autobiography, Mill explicitly says that he became a socialist:

"Our [Mill and Harriet Taylor] ideal of ultimate improvement went far beyond Democracy, and would class us decidedly under the general designation of Socialists. While we repudiated with the greatest energy that tyranny of society over the individual which most Socialistic systems are supposed to involve. we yet looked forward to a time when society will no longer be divided into the idle and the industrious; when the rule that they who do not work shall not eat, will be applied not to paupers only, but impartially to all; when the division of the produce of labour, instead of depending, as in so great a degree it now does, on the accident of birth, will be made by concert, on an acknowledged principle of justice; and when it will no longer either be. or be thought to be. impossible for human beings to exert themselves strenuously in procuring benefits which are not to be exclusively their own, but to be shared with the society they belong to." -- J. S. Mill

Mill is the canonical example of a classical liberal. His short book On Liberty is still read. How can he also be a socialist?

Reference

• Helen McCabe. 2021. John Stuart Mill, Socialist. McGill-Queen's University Press.
• Matthew McManus. 2024. The Political Theory of Liberal Socialism. Routledge.
• Michael Young. 1958. The Rise Of The Meritocracy: 1870-2033.. Pelican.

Rent As Joint Production

I here think about Sraffa's price equations for rent. I try to formulate them explicitly as the system of price equations in the chapter on joint production. I suppose this is a case of me setting out something that many take as obvious.

The data for Sraffa's price equations include the specification of a number of processes producing commodities. Suppose the l process is operated by a capitalist farmer. And they pay rent on the uth type of land. The price equation is:

(p₁ a_1,l + p₂ a_2,l + ... + p_n a_n,l)(1 + r) + rho_u c_u,l + w a_0,l = p_l

Above, rho_u is rent per acre, and c_u,l are the acres of land per unit output for this process. I hope the remaining symbols are obvious.

Let f_u be the purchase price of an acre of land of the uth type. The above equation can be rewritten as:

(p₁ a_1,l + p₂ a_2,l + ... + p_n a_n,l + f_u c_u,l)(1 + r) + w a_0,l = p_l + f_u c_u,l

where:

rho_u = f_u r

The above equality is equivalent to an infinite sum:

f_u = rho_u/(1 + r) + rho_u/(1 + r)² + rho_u/(1 + r)³ + ...

In an explicit formulation of a model of rent as one of joint production, no elements of the input and output matrices, as seen in the price system, are negative. The price of an acre of land is the present value of the stream of rent payments expected to be received on that land in the future.

The above works for intensive rent. One of the lands has a price of zero in the case of extensive rent. How is that expressed in the price equations? I suppose only commodities, that is, goods with positive prices, enter into the equations. Air, being free, does not appear in the price equation for some chemical process that requires nitrogen in some way, for example. But then the solution to a problem of the choice of technique is taken as given. I have a difficulty with many of the chapters in the part on joint production in Sraffa's book.

Textbooks For Marxian Political Economy

Suppose you find Marx's Capital intimidating. You could start with textbooks. I confine myself to a selection in English.

I start with Soviet textbooks. I suppose the Dictionary is not really a textbook. But Apparently, it was a standard reference work. Here are some Soviet textbooks:

• N. Buharin & E. Preobrazhensky. 1922. The ABC of Communism. The Communist Party of Great Britain.
• I. Lapidus & K. Ostrovityanov. 1929. An Outline of Political Economy: Political Economy and Soviet Economics. Martin Lawrence.
• Institute of Economics of the Academy of sciences of the USSR. 1954, 1957. Political Economy. London: Lawrence & Wishart.
• I. Frolov (ed.) 1967, 1984. Dictionary of Philosophy. Moscow: Progress Publishers.

Here are some textbooks:

• Paul M. Sweezy. 1942. The Theory of Capitalist Development: Principles of Marxian Political Economy. Dennis Dobson Ltd.
• Meghnad Desai. 1979. Marxian Economics. Toronto: Rowman & Littlefield.
• Robert Paul Wolff. 1984. Understanding Marx: A Reconstruction and Critique of Capital. Princeton University Press.
• Duncan K. Foley. 1986. Understanding Capital: Marx's Economic Theory. Harvard Univ ersity Press.
• Bob Milward. 2000. Marxian Political Economy: Theory, History, and Contemporary Relevance. Palgrave.
• David F. Ruccio. 2022. Marxian Economics: An Introduction. Polity.
• Deepankar Basu. 2023. The Logic of Capital: An Introduction to Marxist Economic Theory. Cambridge University Press.

I suppose I could expand the above list with reading guides, from David Harvey or Michael Heinrich, for example. The boundary between a textbook and an interpretation is unclear, where by the latter I mean books intendeded to argue with the literature. And I could also have introductory books that are definitely not textbooks, like Eagleton's Why Marx was Right. Even so, I expect this to only be a starting list.

Different authors have different takes. If you want to start with an introduction, I suggest you only pick one.

Some Literature On The Compatibility Of Exhaustible Resources With Classical Political Economy

I have been writing about natural resources that emerge unchanged from the production processes. In contrast, Bidard & Erreygers (2001, 2020) developed the corn-guano model to examine the compatibility of classical political economy with exhaustible natural resources. They argue that a royalty for an exhaustible natural resource will vary over time, in accordance with the Hotelling rule. Parrinello (2004) and Kurz & Salvadori (2011, 2015) argue that when the resource will be exhausted is not well-enough known for the Hotelling rule to apply. The price system, under their assumptions, has enough equations to determine the royalty, without prices varying over time. For Ravagnani (2008), the royalty for an exhaustible resource provides another degree of freedom and is set as a percentage of production by conventions and social norms, much like the natural wage in Ricardo and Marx. Hahnel (2017) emphasizes throughput efficiency, which is related to the rates at which the natural environment’s stock of natural resources and environmental sinks are used by human economic activity. These applications of classical political economy to environmental concerns are outside the scope of of my treatment of absolute, extensive, and intensive rent.

References

• Bidard, Christian, and Guido Erreygers. 2001. The corn-guano model. Metroeconomica, 52(3): 243-253.
• Bidard, C. and G. Erreygers. 2020. Exhaustible resources and classical theory. Oeconomia: History, Methodology, Philosophy, 10 (3): 419-446.
• Hahnel, Robin. 2017. Radical Political Economy: Sraffa versus Marx. New York: Routledge.
• Kurz, Heinz D. and Neri Salvadori. 2011. Exhaustible resources: Rents, profits, royalties and prices. In Volker Caspari (ed.), The Evolution of Economic Theory: Essays in Honour of Bertram Schefold. London: Routledge, 39-52.
• Kurz, Heinz D., and Neri Salvadori. 2015. The ‘Classical’ approach to exhaustible resources: Parrinello and the others. In Heinz D. Kurz and Neri Salvadori, Revisiting Classical Economics. Studies in Long Period Analysis, 304-316. London: Routledge.
• Parrinello, Sergio. 2004. The notion of effectual supply and the theory of normal prices with exhaustible resources. Economic Systems Research, 16(3): 311-322.
• Ravagnani, Fabio. 2008. Classical theory and exhaustible natural resources: notes on the current debate. Review of Political Economy, 20(1).

Sam Tanenhaus Wrong On Buckley On Yale Economics

I have not even yet got to the founding of National Review in this no-doubt authoritative biography. But:

"With the assistance of [Lucille Cardin] Crain and her academic consultants Bill drew up his own list of dangerous books, most of them recently published. First on the list was Paul Samuelson's Economics: An Introductory Analysis. Published in 1948, it laid out the basics of Keynesian and neoclassical theories and in the next years became the most widely used textbook in the field. Its arguments contradicted the laissez-faire ideas to which many Yale donors and alumni subscribed. And, Buckley strongly suspected, those donors and alumni might not be aware that 'the net influence of Yale economics [is] thoroughly collectivistic.'" -- Sam Tanenhaus, 2025. Buckley: The Life and Revolution that Changed America. New York: Random House (204).

I suppose this could be a matter of judgement. I say that, as far as economics goes, Lorie Tarshis' textbook was first on the list. William F. Buckley was participating in an extra-academic intervention so successful that most economists are probably not even aware that Tarshis' textbook existed.

A Paradox For Those Who Know Mathematics

Bri the Math Guy on What to Know for Real Analysis

If you know mathematics, you will be quite willing to deny that.

Some who know me think that I am good with mathematics. Apparently, I used to say quite frequently, when trying to explain something, "It is just math."

I suspect that a lot of those exposed to some advanced mathematics will disclaim knowledge of mathematics. We all have had the experience of taking a couple of hours of reading one or two pages of math and still not being sure of our understanding. Perhaps we read with paper and pen in hand - I am quite willing to cross things out, instead of erase - and work out each step. And still, after going through an entire proof, not being sure of the point.

Even after you have learned or been exposed to quite a bit, you may find that parts of mathematics whose very existence is surprising. For me, these parts still include model theory and alegbraic geometry.

Some cannot read a book if a single equation appears in it. I know that that is so, but I find hard to grasp. Some math books will say that the only prerequisite is mathematical maturity. And I do recall one teacher suggesting that, for our first go-through, we should read a chapter of our textbook like a novel. One thing I have not come across literature on is how authors of math books decide on the scope.

Anyways, to have a hope at learning some math, you must be willing to put time in, while feeling lost. But maybe I am generalizing too much from my own experience.

As I am writing up my example with absolute, extensive, and intensive rent, I want to NOT present a general model. I'd like to say just see Kurz & Salvadori (1995). But they do not have a general model combining intensive and extensive rent, as I understand it. Nor they have a general model with markup pricing.

So instead of doing math, I am writing about math.

Reference

• Lara Alcock. 2013. How to Study as a Mathematics Major. Oxford University Press

Why Can't We All Get Along?

Little Stephen and the Disciples of Soul

I guess this is current events.

I never was of the right temperament for this idea. But many decades ago, I could be intellectually convinced by those saying that we all want the same things. We just disagree on the means. We can discuss and argue about those means, with some approach to rational arguments.

I was wrong.

In the United States today, many do not want general prosperity for all. They want hierarchy, with themselves on top. They do not want to succeed, unless some others fail. They deny the right to exist for many.

In what kind of society can we live freely with such reactionaries? Many would say the answer is liberal society. We want a minimum set of rules, and a lack of restrictions, such that everybody is free to pursue their own idea of a good life. Many of these rules are like conventions on which side of the road to drive. No ethical question arise here. But if we all agree to drive on the right, each of us can make our own plans to get from one place to another with some confidence that they will succeed.

Questions obviously exist about the content and extent of these rules. I think an extreme degree of inequality, with those towards the bottom having only the prospect of a precarious life, is incompatible with a liberal society.

But what happens when those who do not accept the existence of those who differ too much from them take advantage of the liberal norms? Karl Popper reads Plato's Republic as putting forth the paradox of tolerance:

"Less well known is the paradox of tolerance: unlimited tolerance must lead to the disappearance of tolerance. If we extend unlimited tolerance even to those who are intolerant, if we are not prepared to defend a tolerant society against the onslaught of the intolerant, then the tolerant will be destroyed, and tolerance with them. - In this formulation, I do not imply, for instance, that we should always suppress the utterance of intolerant philosophies; as long as we can counter them by rational argument and keep them in check by public opinion, suppression would certainly be most unwise. But we should claim the right to suppress them if necessary even by force; for it may easily turn out that they are not prepared to meet us on the level of rational argument, but begin by denouncing all argument; they may forbid their followers to listen to rational argument, because it is deceptive, and teach them to answer arguments by the use of their fists or pistols. We should therefore claim, in the name of tolerance, the right not to tolerate the intolerant. We should claim that any movement preaching intolerance places itself outside the law, and we should consider incitement to intolerance and persecution as criminal, in the same way as we should consider incitement to murder, or to kidnapping, or to the revival of the slave trade, as criminal." -- Karl Popper, The Open Society and Its Enemies: Volume 1, footnote 4 to chapter 7.

I'd rather not live in a time where this paradox was directly applicable.

Absolute Rent, Extensive Rent, And Intensive Rent

Figure 1: Rent Curves With and Without Competitive Markets

This post is a continuation of the example in this and this post. It is the first example, in the tradition building on Sraffa, of a numerical example combines absolute, intensive, and extensive rent.

The analysis in those posts can be extended to apply to non-competitive markets. Assume that the rate of profits is s₁ r in the price equation for the process producing iron and s₂ r in any process for producing corn. I now call r the scale factor for the rate of profits. As a normalization, let the sum of s₁ and s₂ be unity. These coefficients are given parameters. They express the existence of persistent barriers to entry between industry and agriculture. For concreteness, I take s₁ ≈ 0.0208506. This parameter expresses a case of the reswitching of the order of rentability, with agriculture obtaining a higher rate of profits than industry.

Capitalists in agriculture have market power in this post. The changes in rent, including increases, is not the result of market power by landlords. The increased market power of farmers changes wage and rent curves. The specific parameters for markups considered here eliminate the reswitching of techniques and capital-reversing, at any level of net output. Otherwise, the capital-intensity of industries is not analyzed in this post. For Marx, absolute rent is created from more surplus value being generated in agriculture, given its low organic composition of capital. This additional surplus value is not shared in a common pool because of the market power of the class of landlords. This article neither investigates nor justifies Marx's specific mechanism. Nevertheless, I identify the differences in rent brought about relative market power among capitalists in different economic sectors with absolute rent.

The price systems for each technique are altered by the differences in relative markups between industry and agriculture. The variation of the feasibility of techniques with net output is independent of prices. Table 3, in this post, applies to this numerical example of a model of non-competitive markets. Omicron, Rho, Tau, and Omega remain feasible at the highest level of net output. Figure 2 presents the wage curves for this example of non-competitive markets, and Figure 3 is a detail.

Figure 2: Wage Curves With Non-Competitive Markets

Figure 3: Wage Curves With Non-Competitive Markets (Detail)

With these specific values for relative markups, Omicron and Rho are each cost-minimizing for a range of the scale factor for the rate of profits when net output is towards its maximum (Table 5). Type 1 land is not scarce and pays no rent when Omicron is cost-minimizing. Type 2 land is not scarce under Rho. At the highest level of net output, only extensive rent is obtained. No intensive rent is paid, whatever value the scale factor for the rate of profits takes on. The orders of efficiency and rentability match and do not vary with the scale factor for the rate of profits when Omicron is cost-minimizing. The order of efficiency varies, when Rho is cost-minimizing. Rho exhibits the reswitching of the order of rentability.

Table 1: Cost-Minimizing Technique

Range	Technique	Order of Efficiency	Order of Rentability
0≤r≤9.5%	Omicron	3,2,1	3,2,1
9.5≤r≤41.7%	Rho	3,1,2	3,1,2
41.7≤r≤87.8%		1,3,2
87.8≤r≤114.9%			1,3,2
114.9≤r≤122.0%			3,1,2

Figure 1, at the top of this post, graphs the rent curves for the example. Both the wage and rent per acre are functions of the (scale factor for the) rate of profits. The analysis of the choice of technique would be different if the wage were taken as given exogenously. In the case in this post, rent per acre is less under Omicron when the capitalists in agriculture have more market power. The variation in rent per acre with increased market power for farmers otherwise reflects the change in the cost-minimizing technique. Landlords who own Type 1 land obtain greater rent from the increased market power for agriculture in the example. The same is true for landlords who own Type 3 land, expect for low rates of profits. Landlords who own Type 2 land, on the other hand, are better off with competitive markets.

This level of increased market power for the capitalists in agriculture alters the analysis the choice of technique at every level of net output (Figure 4). Consider the range of the scale factor for the rate of profits at which Delta is cost-minimizing. As output expands, at the bottom of this range, capitalists choose to operate processes IV and V on Type 3 land to expand output. When output can no longer be expanded by extending cultivation with process IV, they bring Type 2 land under cultivation, and then, with the adoption of Omicron, Type 1 land enters into the mix. At a slightly higher scale factor, the capitalists cultivate Type 1 land after Type 3 land is completely farmed with process IV. Ultimately, under Rho, they also cultivate Type 2 land. Consider a still higher scale factor, but still within the range where Delta is initially cost-minimizing. The capitalists first expand output, when Type 3 land is fully cultivated, by starting to farm Type 1 land. They operate processes IV and V side-by-side on Type 3 land only after Type 1 land has become scarce.

Figure 4: Cost-Minimizing Techniques With Non-Competitive Markets

In all of these cases, and for even a higher scale factor, process IV is ultimately operated on Type 3 land alone, and only extensive rent is obtained. Yet, with intensive rent being obtained somewhere along the expansion of output, process V enters into the determination of the order of efficiency. The switch points between the wage curves for Alpha and Gamma and between Alpha and Phi are irrelevant to the determination of the order of efficiency when Rho is cost-minimizing. The order of efficiency varies, though, around the switch point between Alpha and Delta. As with competitive markets, the existence of intensive rent as output expands affects the order of efficiency under extensive rent at the highest level of output.

Ben Shapiro Being Stupid About Marx

Shapiro Explaining Critical Race Theory Correctly Before Criticizing It

I have not actually read Shapiro's Lions and Scavengers" The True Story of America (and Her Critics). I rely on a screenshot in a tweet from Matt McManus.

The above video shows that Shapiro recognizes that you should accurately set out a position before criticizing it. He also gets to boast about his law degree. (Those wanting to laugh at Shapiro probably prefer this interview with Andrew Neil on the BBC.) Anyways, here is a Shapiro take on Karl Marx:

"Marx heavily relied on the labor theory of value. He posited that if you could calculate the value of anything objectively, then anything beyond that value had to be a surplus - an unnecessary bit of profiteering added by the person selling a good, product, or service. So, if the labor value of a potato was, say, $2 - the labor put into growing the potato, harvesting it, and selling it to the grocer - and the grocer sold it for $3 to an end customer, the grocer was adding on an unfair surcharge of $1, thus increasing the price falsely so as to extract profit. Marx said the world could be made fairer and better by preventing the grocer from extracting profit, thus lowering prices and making potatoes more affordable. The grocer was actually exploiting his customers when he 'earned' a profit.

The answer, said Marx, was to set prices from the top down. Simply set the price of potatoes at $2, remove the grocer’s evil profit, and make everything more plentiful and cheaper." – Ben Shapiro. Lions & Scavengers: The Tue Story of America.

The above is so far from anything Marx wrote that I do not think any comment is needed. Nevertheless, I will go on.

Marx does not explain profits as the result of a seller tacking a surcharge on the price of a commodity, where that price is the labor value of the commodity. In fact, he specifically rejects that idea.

For purposes of exposition, Marx assumes, in volume 1 of Cqpital, that prices tend towards labor values, under the capitalist mode of production. He wants to explain the origin of profit under the assumption that prices are 'fair' that many pro-capitalists and his predecessors put forth shortly before.

Surplus value, the source of profits, are obtained by capitalists not paying out the full value added by labor as wages to workers. The distinction between labor-power and labor is central to Marx's explanation. Labor-power is the ability to work under the direction of the agents of capital.

The suggestion that Marx thinks profit is only obtained by the merchant selling a commodity to the consumer is another absurdity. For Marx, surplus value is obtained at every step of production. The farmers hiring workers to plant, grow, and harvest potatoes exploits their workers. Shapiro can only be joking when he suggests that Marx thinks merchants exploit customers.

The final bit of silliness I want to highlight is that Marx argues a post-capitalist economy should set prices top-down to labor values. Marx is descriptive in Capital. He does not provide a blueprint for post-capitalist societies. He mocks the idea in the afterword to the second German edition of volume 1. In The Poverty of Philosophy, he specifically attacks Prodhon's idea of setting prices to labor values. Anarchists might argue that Marx is inconsistent and draws on Proudhon's ideas in his letter, Critique of the Gotha Programme. But any use of labor values in socialism differs from Marx's theory in Capital. Furthermore, many would argue that Marx's concept of value does not apply to communism, for example.

Apparently, Shapiro has been spouting nonsense about Karl Marx for years. Ben Burgis once noticed.

Update: Ben Burgis has a substack article about the above. And a YouTube video discussion with Stefan Bertram Lee.

Matt McManus and Nathan J. Robinson give the book a negative review.

Shapiro's book has a lot more wrong with it.