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.

Variation Of Cost-Minimizing Technique With Increased Net Output And Extensive And Intensive Rent

Figure 1: Cost-Minimizing Techniques

1.0 Introduction

This post is an extension of one of my examples with extensive and intensive rent. I do not expect this to make sense unless you have read that post. I struggle to easily visualize details of that example.

David Ricardo distinguishes between extensive and intensive rent. He seems to put more emphasis on extensive rent. Even so, he considers the combination of extensive and intensive rent:

"It often, and, indeed, commonly happens, that before No. 2, 3, 4, or 5, or the inferior lands are cultivated, capital can be employed more productively on those lands which are already in cultivation. It may perhaps be found, that by doubling the original capital employed on No. 1, though the produce will not be doubled, will not be increased by 100 quarters, it may be increased by eighty-five quarters, and that this quantity exceeds what could be obtained by employing the same capital, on land No. 3." - David Ricardo, Principles

Ricardo does not recognize that the order of efficiency, also known as the order of fertility, differs from the order of rentability. Nor does he recognize that these orders depend on the distribution between workers and capitalists. Nevertheless, analysis of rent by Ricardo and his contemporaries is the starting point for many interesting analyses.

2.0 Some Repetition

Assume that net output in the example consists of a numerically identical number of tons iron and bushels corn. After different levels of net output, a different set of techniques are feasible. Table 2 here defines the techniques, and Table 3 specifies which techniques are feasible as net output expands.

I think I want to repeat at least the wage curves for the example. Accordingly, Figure 2 plots the wage curves for the cost-minimizing techniques, when net output is almost as large as it can be. Figure 3 is an enlargement. The figures also show the wage curve for Delta.

Figure 2: Wage Curves for Feasible Techniques

Figure 3: Wage Curves for Feasible Techniques (Detail)

3.0 Cost-Minimizing Techniques

Figure 1, at the top of the post, depicts which techniques are cost-minimizing at which rate of profits, as net output expands. The dashed lines are points at which the order of efficiency changes, while the cost-minimizing technique does not change. They are intersections of wage curves above the wage curves for the cost-minimizing technique. The dotted lines are points at which the order of rentability changes. They are intersections of rent curves for the cost minimizing technique. Figure 4 is an enlargement.

Figure 4: Cost-Minimizing Techniques (Detail)

3.1 Extensive Rent Alone

The range of the rate of profits for which Tau is cost-minimizing, when net output is near its maximum, illustrates the analysis of extensive rent. The order of efficiency can be calculated by working downwards from the wage curves in this range. Type 3 land is partially farmed under Tau, using the same process on type 3 land as in the Gamma process. The wage curves for Alpha and Beta intersect at approximately 48 percent. This switch point corresponds to a change in the order of efficiency. When Alpha is cost-minimizing, at the lowest output region for net output, the order of efficiency for Tau is Type 1, 2, and 3 lands. When Beta is cost-minimizing, the order of efficiency is type 2, 1, and 3 lands.

The order of rentability does not vary in the range in which Tau is cost-minimizing. The rent curves for Tau do not intersect. Consequenty the orders of efficiency and rentability differ in part of the range for which Tau is cost-minimizing. Type 1 land can obtain greater rent per acre, even though type 2 land is more fertile at the given range of the rate of profits. These results echo the analysis of extensive rent in Alberto Quadrio Curzio (1980).

When net output is small enough such that Alpha and Beta are feasible, the switch point at which the order of efficiency changes for Tau exhibits capital-reversing. A higher wage is associated with firms hiring more workers, given net output. But this intersection of the Alpha and Beta wage curves is not associated with any real Wicksell effects for Tau. Quantity flows do not vary around the switch point when Tau is cost-minimizing. Both above and below the switch point, Type 1 and 2 lands are fully farmed, with the same processes be operated on each land.

Consider regions of net output for which Epsilon and Theta are feasible. Theta requires a greater input of labor for a given net output. So here too, this switch point exhibits capital-reversing, while arising in an example with extensive rent.

3.2 Extensive and Intensive Rent

The range of the rate of profits for which Omicron or Omega are cost-minimizing illustrates effects of a combination of extensive and intensive rent. Process V is operated on type 3 land for the Phi, Kappa, Chi, Pi, Xi, and Upsilon techniques. On the other hand, process IV is operated on type 3 lands for Type 3 land for the Iota and Omicron techniques. Even though process V is not operated, when net output is towards its maximum, in the range of the rate of profits where Omicron is cost-minimizing, process V contributes towards determining the order of efficiency. The switch point between Beta and Gamma has no effect on the order of efficiency. A combination of intensive and extensive rent is needed to see the possibility of non-operated processes impacting the order of efficiency.

When the Omega technique is cost-minimizing, both processes are operated on Type 3 land. Nevertheless, the switch point between Alpha and Gamma has no effect on the order of efficiency in this range.

The order of efficiency happens to match the order of rentability when Omicron is cost-minimizing. The rent curves for Omega intersect three times in the range of the rate of profits is cost-minimizing. And the order of efficiency also changes. Sometimes these orders diverge when Omega is cost-minimizing. In examples with both intensive and extensive rents, lands that are less fertile at a given rate of profits can obtain greater rents per acre.

Marginalism is a generalization of Ricardo's theory of extensive rent and its application to all factors of production. Perhaps this mismatch of the orders of efficiency and rentability is another indication of how the generalization is unwarranted.

4.0 Conclusion

Alberto Quadrio Curzio put a lot of work into investigating the theory of rent in Sraffa's approach to political economy. This post can be said to be a clarification of this comment:

"Even when the intensive rent disappears, the effects of intensive cultivation persist in the different productive processes applied to the different lands, which affect the extensive differential rents." - Alberto Quadrio Curzio & Fausta Pellizzari (2010: 46)

The process on type 3 land that goes into the system of equations for Omicron differs from the process on type 3 land used in calculating the order of efficiency. This contrast reflects the phenomenon of intensive rent, even though no intensive rent is obtained when Omicron is cost-minimizing.

The example emphasizes the need to consider technical change. Net output cannot be increased after a hard limit, imposed by land-like natural resources. Increased production is only possible with the introduction of new processes and techniques or decreases in some coeficients of production for existing processes.

The most surprising finding, for me, is that types of lands cannot generally be ordered by efficiency (also known as fertility), from most fertile to least, by observing the processes operated on the lands at a given rate of profits. This result is in tension with Sraffa's methodology. Sraffa had an aversion to counterfactuals (Sen 2003). He wanted to not not use data - for example, on processes available in the technology but not currently operated - that cannot be obtained by the 'man from the moon' (Kurz & Salvadori 2004) or in a snapshot of an ongoing economy (Roncaglia 1975).

Otto Neurath And Happiness

I think of Otto Neurath primarily as a member of the Vienna circle. Rudolf Carnap and Moritz Schlick were two ther prominent members. They developed the philosophy of logical empiricalism or logical positivism. No woolly dialectics or cultural criticism for them.

It was a custom, when they were meeting at their preferred coffee-house in 1920s Vienna, to interrupt any speaker who had strayed into much-despised metaphysics. Neurath did this so often that they told him to hold up his hand whenever the speaker said something that was not metaphysics.

When Neurath fled the Nazis, he took his wife and his mistress with him. They got along well.

A number of revolutions, inspired by the Bolsheviks, convulsed central Europe after World War I. I think of Hungary and the Spartacists, in particular. Neurath began implementing his ideas for central planning under the Bavarian Soviet Republic. He advocated planning in kind, without monetary prices.

This was long before the definition of the Gross Domestic Product (GDP). But Neurath would not be interested in optimizing GDP in his planning. He did not want to duplicate capitalism. Rather he looked at a broader array of measures:

"Neurath ... pioneered a measure of living standard. He took variables that are now familiar to economists, such as nutrition, health, life expectancy, housing, clothing, incidence of crime. He was also concerned to build these up in a single measure, and its level, as well as its distribution, were to be the concern of the socialist planner." – Meghnad Desai. 2002. Marx's Revenge: The Resurgence of Capitalism and the Death of Statist Socialism.

This measure is like the United Nations Human Development Index (HDI), Bhutan's Gross National Happiness (GNP), or, say, metrics promoted by Joseph Stiglitz.

I suppose I ought to mention ISOTYPE, Neurath's pictorial language or symbols.

A century has shown that an unregulated, unrestrained capitalism does not deliver a broad-based prosperity for all, in which most can fully develop their capabilities. Suppose you care about the majority of your fellow citizens, not just a few at the top. Empirically, you should vote for socialists.

Sraffa On The Use Of The Notion Of Surplus Value

Sraffa, in his archives in the 1940s and 1950s, is quite appreciative of Karl Marx's analysis of capitalism. This appreciation contrasts with the opinion embodied in the label 'neo-Ricardian', which Bob Rowthorn invented.

I know about the passages below in the Sraffa archives from Riccardo Bellofiore. The archivist, Jonathan Smith, has dated this entry from 1955-1959, late in the writing of Production of Commodities.

I do not want to focus on whether Marx or Sraffa are correct or not. I would want to work out a simple example. Besides, Sraffa seems not convinced of how to analyze the reduction in the working day, when starting at prices.

But I want to note that Sraffa is very much using Marxist concepts: vulgar economics, labor values, prices of production, surplus value, exploitation, and rates of exploitation. And the analysis is based on Marx. Surplus value comes from extending the working day past the point at which workers reproduce their labor power.

"Use of the Notion of Surplus Value

"The prolongation of the working day beyond the point at which the labourer would have produced just an equivalent for the value of his labour-power ..." (Cap., Engels transl. p. 518) cp p. 539 [Chapter Sixteen: Absolute and Relative Surplus-Value]

Put it the other way round. If starting from capitalist society the working day is shortened till there is no surplus value left, this shortening must be equal for all: if it is, the prices of the commodities will change [owing to change in the rate of profits, which vanishes], but the wages will remain unchanged : if it is not, and the working day is reduced to the extent of the profits made in each industry, then prices would remain unchanged* after the shortening [for the number of (shorter) labor days, in industries having a high organic composition of capital, would increase in the same proportion as the fall of profits] but wages would be different.

[Footnote:] *(28.12.41) But profits would be different (after the reduction) in different industries!

[Marginal note:] c/p Letters of M and E 129-32 (letter of M. 2.8.62)

In other words, if we start from profits (as vulgar economy does) we reach the conclusion that the rate of exploitation is different in different industries, being higher in the more highly capitalised ones – which is not [and indeed contrary to] the fact. If we start from surplus value, which is equal in all industries, we get the correct measure of exploitation. The former conclusion is patent nonsense, and no view of exploitation could be based on it.

Note that the former (profits) goes with a theory of prices, the latter, of value (as defined below).

12.11.40 [Price is an exchange ratio which equalises rates of profit on capitals. Value is an exchange ratio which equalises rates of surplus-value on labour. If commodities exchanged at their values, profits would be different for different capitals, and capitals would move: therefore, this competition of capitals causes them to exchange at their prices.

The question is: are the rates of exploitation different? and if so why doesn’t labor move, and restore values and equality of rates of surplus value?]

The starting point is "the prolongation of the working day beyond the point at which the labourer would have produced just an equivalent for the value of his labour power" (Cap., Engels Tr. 518)

This point cannot be determined without reference to the value of the product (unless the labourer produces himself all the commodities he consumes).

But the point varies if we take value and if we take price.

Now, we are comparing the actual state with a hypothetical one in which only the necessary labour is performed.

In the actual state commodities are exchanged at their prices, whilst in the hypothetical state (where there would be nothing to be paid out in profits) at their values.

Which scale should we adopt for both states, in comparing them? It may be said: neither – each state has its own scale and that only is appropriate to it. No comparison can be made directly between the two extreme states. [marginal note says, "wrong, see p. 56.] We cannot imagine to move gradually from the actual state, shortening the working day; as we start from the actual state, we use its own scale, i.e. prices, in determining the ultimate goal towards which we move [and that will give different reductions for different branches of industry; but as we pass to successive other states, with shorter and shorter working days, the scale to be used changes, and prices move nearer (as the rate of profits is reduced) to values – so does the "point" aimed at change; until, on the threshold of the state in which only the necessary labour is performed, the prices will practically coincide with values, and the point aimed at with that determined by the scale of values, i.e. all labourers will have had their hours reduced in the same proportion. [The converse is true: if starting from the hypothetical state we prolong the working day by this method, we reach the actual state, having prolonged it for all labourers proportionally/equally, but through the change in prices having raise the profits in each branch proportionally with its capital]

Note that if we had adopted straightway values, and made the comparison between the two extreme cases, we should have obtained the same, correct, result. But if we had adopted prices, and made that comparison, it would have led us astray: the 'point' indicated by prices [i.e. different reductions in different industries] would have been false when the hyp. state was reached – for on the basis of values some labourers would be working more, and some less, than the necessary hours.

The imaginary process (described above, p. 3 bottom) of gradually shortening the working day, on the basis of the prices appropriate to each intermediate point, and therefore in different proportions for different industries, requires further consideration. As it stands, it is only correct at the wo extreme points [or rather only at the final point], but false at all intermediate ones: for, e.g., on the first step, when the rate of profit is reduced from 6 to 5%, the day of every worker must be reduced in one and the same proportion, and not in different proportions: it is clear that the latter method would give immediately absurd results.

In fact, this shows that the way in which I have argued the point on p. 1 is wrong (too weak). The objection of the vulgar economist is that the surplus produced in each industry (or firm) is measured by its profits. If he agreed to call it exploitation he would say that this is higher (absorbs a larger proportion of the working day) in the industries having more capital per worker. Therefore, he would have to conclude that, if exploitation has to be reduced in the different industries in such proportions as would maintain the rate of profit equal between them, at the lower level, this would require a larger reduction of the working day in the more capitalised industries. It can be shown, by a numerical example, that this is not the case. That on the contrary if the was reduced equally in all industries, the rate of profits would also be reduced equally. This result is made possible by a simultaneous change in prices – those of highly capitalised industries falling (when the rate of profits falls) relatively to the other prices. So that the larger fall in the surplus of such industries has two sources: a) the reduction in working way (common to other industries), (b) the fall in the relative price of their product (peculiar to the highly cap. industries)

29.12.41 [A third source, working in the same direction, would at first sight appear to be the increased depreciation allowances for capital, as the rate of interest falls. However this is a delusion: the 10 loom case shows that it is constant. That is to say, it is constant if real capital has to be maintained intact, though allowing value of semi-used capital to fall as rate of interest falls - and this is the relevant case. Its rise only if money value has to maintained as originally, in case

29.12.41 The previous paragraph is misleading. There is a third source, even if depreciation allowances are regarded as constant. For the value (or rather, in M's sense, the "price") of capital goods falls with the fall in the rate of interest. Therefore, when the rate of s.v. falls, the profits of industries with a large amount of capital per man fall still more owing to this third source - even if the capitals are no more (but no less!) "durable" than in other industries (if they are more durable in the highly capitalised industries than in others, this is a fourth source - for with fall in interest the more durable capitals falls more than that of the less durable ones).

[The whole subject of the "measure of capital" requires investigation in this connection. It has a striking similarity to the contradiction of "prices" and "value" of commodities, and it also depends on the equalisation of the rate of profits. One should start from the "value" of capitals (i.e. quantity of labour necessary to construct them) and see how the requirement of equal rates of profits leads to "prices" of capitals different from their "values".]

30.12.41 This business of 3 or 4 sources is wrong. There are only two sources: a) the reduction of working time in the industry, which reduces the quantity of goods produced; b) The reduction in the price at which the product is sold. The fall in the value of capital of certain industries along with the fall in the general rate of profits and other possible causes, contribute to source b, but don't add anything besides b.

[N.B. The fact that the value of capital (and therefore its "quantity" or magnitude) varies with the rate of profits (and generally cannot be known without knowing prices and rate of profits) makes nonsense of many cornerstones: 1) "Sacrifice of waiting", but how if they don’t know what they are abstaining from? 2) rate of interest, or marg. prod. of cap., as criterion for distribution of resources; but how, if the same resource (in "value") becomes larger or smaller (in "price") according as it used in one way or another?

5.1.42 Those who regard Marx's transition from values to prices, by the necessity of equalising the rate of profit, as a trick, should say the same of Ricardo's (and the whole marginal school) method of determining cost of production by considering only that on the marginal land, by the necessity of equalising the price of all bushels of corn, on whichever land they may be procured. Cannan does so (Rev. of Ec. Theory, p. 178): Ricardo 'did the trick by little more than an arbitrary exercise of the right to define terms ..." -- Piero Sraffa D3.12.46/57r – 63r

Nothing like the above is in Sraffa's book. Connections to Marx are less apparent, although some reviewers perceived them. Counterfactual reasoning is mostly eschewed. The length of the working day is not discussed, but taken as given.

Sraffa does not seem very confident about whether he should start with value or prices and how he should proceed if he adopts the latter. He does see the importance of what was later called price Wicksell effects. I want to note that the next pages in the archive are a draft of the chapter on land in Sraffa's book.

By the way, Ian Steedman has a chapter towards the end of Marx after Sraffa illustrating the analysis of the length of the working day. Consistent with his general approach, he uses data on physical quantity flows and does not take the point at which prices are values and labor is not exploited as a reference point.

Another Example With A Cost-Minimizing Technique With Intensive And Extensive Rent

Figure 1: Detail on Variation of Rent per Acre with Rate of Profits

1.0 Introduction

I have been exploring examples with both intensive and extensive rent. I try to make this a stand-alone post, with possibilities for later elaborations.

Consider a model of the production of commodities with non-produced means of production that are unchanged by their use in production. In other words, they are types of land. In a simple model of extensive rent, a single agricultural commodity, 'corn', can be produced, on each type of land, with a single production production. This post expands a simple multi-commodity model to postulate the existence of two production processes on one type of land. The model then combines intensive and extensive rent, depending on the choice of technique.

In the example, all three types of land are at least partially cultivated to satisfy requirements for use. Whether or not all three types of land obtain a rent depends on the level of profits. A mixture of intensive and extensive rent is obtained only for a range of the rate of profits.

This example works like a lot of mine. It is of perhaps the minimum structure needed to make my points in a model with more than one commodity produced, with a circular structure of production and labor inputs in all processes. Yet its elaboration seems complicated. And surprising results arise here or there. They would be more impressive if they occurred in a larger range of the rate of profits.

2.0 Technology, Resources, Final Demand, and Feasibility

A model of the production of commodities is specified by the technology, the endowments of unproduced natural resources, and the requirements for use. Technology is specified, in a discrete technology, by coefficients of production for each production process. Each process is assumed to require the same time to complete and to exhibit constant returns to scale, up to the limitation imposed by scarce land. The endowments of each type of land are specifed. Requirements for use are specified by final demand.

Table 1 presents coefficients of production for the example. Two commodities are produced, iron and corn. Aside from the use of land, joint production is not possible. Multiple types of land (that is, three types) exist. Only one agricultural commodity, corn, can be produced on the processes in which land is used. For one type of land, more than one process can be operated on land. Only one process is known for producing iron, the industrial commodity. Each column in Table specifies the person-years of labor, acres of a type of land, tons of iron, and bushels of corn needed to produce a unit output of the specified commodity.

Table 1: The Coefficients of Production

Input	Industry
	Iron	Corn
	I	II	III	IV	V
Labor	1	9/10	3/5	29/50	9/20
Type 1 Land	0	1	0	0	0
Type 2 Land	0	0	49/50	0	0
Type 3 Land	0	0	0	2/5	2
Iron	9/20	1/40	3/2000	29/500	2/30
Corn	2	1/10	9/20	13/100	13/100

Various techniques (Table 2) can be defined with this technology. All twenty-four letters in the Greek alphabet are needed to specify the techniques. Not all techniques are feasible, given technology, endowments, and requirements for use. Land is not scarce for the Alpha, Beta, Gamma, and Delta techniques, and ownership of land obtains no rent. The Epsilon through Upsilon techniques are examples of extensive rent. One type of land obtains a rent in the Epsilon through Xi techniques. All three types are farmed in Omnicro through Upsilon, and two types obtain a rent. Phi is an example of intensive rent. Chi, Psi, and Omega are examples of the combination of intensive and extensive rent.

Table 2: Techniques of Production

Technique	Processes	Land
		Type 1	Type 2	Type 3
Alpha	I, II	Partially farmed	Fallow	Fallow
Beta	I, III	Fallow	Partially farmed	Fallow
Gamma	I, IV	Fallow	Fallow	Partially farmed
Delta	I, V	Fallow	Fallow	Partially farmed
Epsilon	I, II, III	Partially farmed	Fully Farmed	Fallow
Zeta	I, II, IV	Partially farmed	Fallow	Fully Farmed
Eta	I, II, V	Partially farmed	Fallow	Fully Farmed
Theta	I, II, III	Fully Farmed	Partially farmed	Fallow
Iota	I, III, IV	Fallow	Partially farmed	Fully Farmed
Kappa	I, III, V	Fallow	Partially farmed	Fully Farmed
Lambda	I, II, IV	Fully Farmed	Fallow	Partially farmed
Mu	I, III, IV	Fallow	Fully Farmed	Partially farmed
Nu	I, II, V	Fully Farmed	Fallow	Partially farmed
Xi	I, III, V	Fallow	Fully Farmed	Partially farmed
Omnicron	I, II, III, IV	Partially farmed	Fully Farmed	Fully Farmed
Pi	I, II, III, V	Partially farmed	Fully Farmed	Fully Farmed
Rho	I, II, III, IV	Fully Farmed	Partially farmed	Fully Farmed
Sigma	I, II, III, V	Fully Farmed	Partially farmed	Fully Farmed
Tau	I, II, III, IV	Fully Farmed	Fully Farmed	Partially farmed
Upsilon	I, II, III, V	Fully Farmed	Fully Farmed	Partially farmed
Phi	I, IV, V	Fallow	Fallow	Fully Farmed
Chi	I, III, IV, V	Fallow	Fully Farmed	Fully Farmed
Psi	I, II, IV, V	Fully Farmed	Fallow	Fully Farmed
Omega	I, II, III, IV, V	Fully Farmed	Fully Farmed	Fully Farmed

I assume that 100 acres of each of the three types of land are available. Net output consists of 55 tons iron and 55 bushels corn. This completes the specification of the example. The parameters for the example are fairly arbitrary. They are chosen to ensure reswitching of techniques between the Alpha and Beta techniques when net output is small and no land is scarce. The given net output, however, results in all types of land being scarce.

Under these assumptions, Omnicron, Rho, Tau, and Omega are feasible. All three types of land are farmed under these three techniques. Type 1 land is only partially farmed under Omnicron, and it is non-scarce and does not obtain a rent. Type 2 land does not obtain a rent under Rho. Type 3 land does not obtain a rent under Tau. All three types are fully farmed under Omega. A linear combination of processesare IV and V are operated side-by-side under Omega. Type 3 land is therefore scarce under Omega. All three types are farmed under Omnicron, with non-scarce Type 3 land only partially farmed.

3.0 Quantity Flows

What techniques are feasible varies as output expands. I claim that, unlike in models with only extensive rent, the order of efficiency depends on the sequence in which techniques are cost-minimizing as net output expands. So I find the following remark dubious:

"No changes in output and (at any rate in Parts I and Il) no changes in the proportions in which different means of production are used by an industry are considered..." (Sraffa 1960)

Sraffa considers land, along with other examples of joint production, in part II.

If net output is low enough, gross outputs can be such that one type of land is partially farmed. The Alpha, Beta, Gamma, and Delta techniques are all feasible. For the example, the Delta technique is the first technique to become infeasible as output expands. It is replaced by the Eta, Kappa, and Phi techniques, in which type 3 land is scarce and obtains a rent. Table 3 shows which techniques become infeasible or feasible as output expands. Without an improvement in technology, the maximum output for the Omicron, Rho, Tau, and Omega techniques provides a hard limit for this economy.

Table 3: Output Regions

Output Region	Feasible Techniques
1	Alpha, Beta, Gamma, Delta
2	Alpha, Beta, Gamma, Eta, Kappa, Phi
3	Alpha, Gamma, Epsilon, Eta, Kappa, Mu, Xi, Phi
4	Gamma, Epsilon, Eta, Theta, Kappa, Lambda, Mu, Nu, Xi, Phi
5	Gamma, Epsilon, Eta, Theta, Lambda, Mu, Nu, Pi, Phi, Chi
6	Gamma, Epsilon, Theta, Lambda, Mu, Pi, Sigma, Phi, Chi, Psi
7	Gamma, Lambda, Mu, Pi, Sigma, Tau, Upsilon, Phi, Chi, Psi
8	Zeta, Iota, Lambda, Mu, Pi, Sigma, Tau, Upsilon, Chi, Psi
9	Zeta, Iota, Lambda, Mu, Tau, Chi, Psi, Omega
10	Zeta, Lambda, Omicron, Tau, Psi, Omega
11	Omicron, Rho, Tau, Omega

4.0 Prices of Production

A system of equations specify prices of production for each technique. All operated processes pay the same rate of profits. Rents and wages are paid out of the surplus at the end of the year. A type of land that is only partially farmed is not scarce and pays no rent. I take the net output as the numeraire.

As an example, the system of equations in following five displays specify the prices of production for the Omega technique.

(p₁ a_1,1 + p₂ a_2,1)(1 + r)+ w a_0,1 = p₁

(p₁ a_1,2 + p₂ a_2,2)(1 + r) + rho₁ c_1,2 + w a_0,2 = p₂

(p₁ a_1,3 + p₂ a_2,3)(1 + r) + rho₂ c_2,3 + w a_0,3 = p₂

(p₁ a_1,4 + p₂ a_2,4)(1 + r) + rho₃ c_3,4 + w a_0,4 = p₂

(p₁ a_1,5 + p₂ a_2,5)(1 + r) + rho₃ c_3,5 + w a_0,5 = p₂

Prices of production for the other techniques are specified by a subset of the system of equations for the Omega technique. Each operated process corresponds to an equation in the corresponding system of prices of production. The rent on land that is partially farmed is zero in the corresponding equation, since land in excess supply is not scarce.

The numeraire is specified by a further equation, where the column vector d represents net output.

p₁ d₁ + p₂ d₂ = 1

4.1 On the Solution

A linear combination of the last two equations in the system of prices of production, for the Phi, Chi, Psi, and Omega techniques, eliminates the rent of type 3 land. In the techniques with extensive rent, one of the equations for a corn-producing process does not contain a term for rent either.

This equation for a corn-producing process or the linear combination of the last two equations can be combined with the first equation, for the iron-producing process. This results in a system of two equations in four unknowns, the price of iron, the price of corn, the wage, and the rate of profits. The equation for the numeriare removes one degree of freedom. If the rate of profits is taken as given, this is a linear system which can be solved for prices of produced commodities and the wage.

The rent per acre can be found for each equation remaining in the original system of equations for a technique. The Alpha, Epsilon, Zeta, Eta, Omnicron, and Pi techniques, for example, have the same solution for prices of produced commodities and the wage. Epsilon, Omnicron, Pi have the same rent per acre on type 2 land. Zeta and Omnicron have the same rent per acre on Type 3 land, while Eta and Pi have the same rent per acre on Type 3 land.

4.2 Wage and Rent Curves

Given the technique, the wage is therefore a function of the rate of profits. Likewise the rent on lands that are always fully-farmed with that technique is also a function of the rate of profits.

The wage is a declining function of the rate of profits in the first four techniques and in the 16 techniques with extensive rent alone. A maximum wage corresponds to a rate of profits of zero, and a maximum rate of profits corresponds to a wage of zero. The wage curve can be upward-sloping in models of extensive rent. The wage curves, in the example, happen to be downward-sloping in the example. Figure 2 shows the wage curves for the feasible techniques in the example. The wage curve for the Omega technique is only shown for the range in which the rents on all three types of land are zero. The enlargement in Figure 3 shows a range of the rate of profits towards the start of the range at which Omega is cost-minimizing.

Figure 2: Wage Curves for Feasible Techniques

Figure 3: Wage Curves for Feasible Techniques (Detail)

Figure 3 plots rent per acre, as a function of the rate of profits, for the feasible techniques in this post. Figure 1, at the top of this post, is an enlargement of part of the range of the rate of profits. Under Rho, both type 1 and type 3 lands are fully farmed. No range of the rate of profits exist in which scarce land under Rho both receive non-negative rates of profits. Thus, Rho, although feasible, can never be cost-minimizing. Omicron, Omega, and Tau are each uniquely cost-minimizing, other than at switch points, for some range of the rate of profits.

Figure 4: Rent Curves for Feasible Techniques

5.0 Cost-Minimizing Techniques and Orders of Efficiency and Rentability

The cost-minimizing technique, at a given level of net output, varies with the rate of profits. Table 4 specifies the cost-minimizing technique in each output region. The rates of profits shown are only approximate.

Table 4: Cost-Minimizing Technique

Output Region	Range	Technique	Order of Efficiency	Order of Rentability
1	0≤r≤15.8%	Delta	3	N/A
	15.8≤r≤18.1%	Beta	2	N/A
	18.1≤r≤48.2%	Alpha	1	N/A
	48.2≤r≤78.5%	Beta	2	N/A
2	0≤r≤6.2%	Phi	3	3
	6.2≤r≤15.8%	Kappa	3,2	3,2
	15.8≤r≤18.1%	Beta	2	N/A
	18.1≤r≤48.2%	Alpha	1	N/A
	48.2≤r≤78.5%	Beta	2	N/A
3	0≤r≤6.2%	Phi	3	3
	6.2≤r≤15.8%	Kappa	3,2	3,2
	15.8≤r≤16.1%	Xi	2,3	2,3
	16.1≤r≤18.1%	Epsilon	2,1	2,1
	18.1≤r≤48.2%	Alpha	1	N/A
	48.2≤r≤78.5%	Epsilon	2,1	2,1
4	0≤r≤6.2%	Phi	3	3
	6.2≤r≤15.8%	Kappa	3,2	3,2
	15.8≤r≤16.1%	Xi	2,3	2,3
	16.1≤r≤18.1%	Epsilon	2,1	2,1
	18.1≤r≤48.2%	Theta	1,2	1,2
	48.2≤r≤78.5%	Epsilon	2,1	2,1
5, 6	0≤r≤6.2%	Phi	3	3
	6.2≤r≤9.5%	Chi	3,2	3,2
	9.5≤r≤15.4%	Pi	3,2,1	3,2,1
	15.4≤r≤15.8%			2,3,1
	15.8≤r≤16.1%		2,3,1
	16.1≤r≤18.1%	Epsilon	2,1	2,1
	18.1≤r≤48.2%	Theta	1,2	1,2
	48.2≤r≤78.5%	Epsilon	2,1	2,1
7	0≤r≤6.2%	Phi	3	3
	6.2≤r≤9.5%	Chi	3,2	3,2
	9.5≤r≤15.4%	Pi	3,2,1	3,2,1
	15.4≤r≤15.8%			2,3,1
	15.8≤r≤16.1%		2,3,1
	16.1≤r≤16.4%	Upsilon	2,1,3	2,1,3
	16.4≤r≤18.1%			1,2,3
	18.1≤r≤24.4%		1,2,3
	24.4≤r≤48.2%	Tau
	48.2≤r≤50.1%		2,1,3
8	0≤r≤6.2%	Iota	3,2	3,2
	6.2≤r≤9.5%	Chi
	9.5≤r≤15.4%	Pi	3,2,1	3,2,1
	15.4≤r≤15.8%			2,3,1
	15.8≤r≤16.1%		2,3,1
	16.1≤r≤16.4%	Upsilon	2,1,3	2,1,3
	16.4≤r≤18.1%			1,2,3
	18.1≤r≤24.4%		1,2,3
	24.4≤r≤48.2%	Tau
	48.2≤r≤50.1%		2,1,3
9	0≤r≤6.2%	Iota	3,2	3,2
	6.2≤r≤9.5%	Chi
	9.5≤r≤12.2%	Omega	3,2,1	3,2,1
	12.2≤r≤12.5%			3,1,2
	12.5≤r≤12.8%			1,3,2
	12.8≤r≤15.8%			1,2,3
	15.8≤r≤16.1%		2,3,1
	16.1≤r≤18.1%		2,1,3
	18.1≤r≤24.4%		1,2,3
	24.4≤r≤48.2%	Tau
	48.2≤r≤50.1%		2,1,3
10, 11	0≤r≤9.5%	Omicron	3,2,1	3,2,1
	9.5≤r≤12.2%	Omega
	12.2≤r≤12.5%			3,1,2
	12.5≤r≤12.8%			1,3,2
	12.8≤r≤15.8%			1,2,3
	15.8≤r≤16.1%		2,3,1
	16.1≤r≤18.1%		2,1,3
	18.1≤r≤24.4%		1,2,3
	24.4≤r≤48.2%	Tau
	48.2≤r≤50.1%		2,1,3

The range of the rate of profits towards its minimum value illustrates that intensive rent can be transient. When the Delta technique, which produces corn only with process V, becomes infeasible, The technique, Phi in which type 3 land is fully farmed with processes IV and V operating side-by-side, becomes cost-minimizing at a rate of profits of zero and nearby.

For a large enough output, Phi is no longer feasible. The cost-minimizing technique at a rate of profits becomes Iota, a technique with only extensive rent. Type 3 land is fully farmed with process IV, and Type 2 land is partially farmed.

Much else can be said about the sequence in which techniques become cost-minimizing, in various ranges of the rate of profits, as net output expands.

5.1 The Order of Efficiency

The order of efficiency is the order in which techniques are adopted with increasing net output at a given wage or rate of profits. The order of efficiency is also known as the order of fertility. Since the order of efficiency varies with the rate of profits, fertility is not defined solely by physical inputs and outputs.

In models with extensive rent, the order of efficiency can be read off the order of wage curves. The range of the rate of profits at which Tau is cost-minimizing in the example illustrates. The order of efficiency here is the sequence of wage curves from the highest to the wage curve for Gamma. Under Gamma, Type 3 land is partially farmed with process IV, as in the Tau technique.

The order of efficiency for Tau varies at the switch point between the Alpha and Beta wage curves. For the lowest levels of net output, this switch point between Alpha and Beta is an example of capital-reversing. But, for Tau, a change in quantity flows is not even associated with this change in the order of efficiency. Capital-reversing does not occur.

Consider the range of the rate of profits between approximately 6.18 and 9.54 percent. As net output expands, the Delta, Kappa, Chi, and Omicron techniques succeed one another. Delta and Kappa operate process V on type 3 land. Chi operates a linear combination of processes IV and V. The sequence of wage curves in this range varies from Delta, Gamma, Beta, Alpha through Delta, Beta, Gamma, Alpha. Yet the order of efficiency does not vary. Even though process IV is operated on type 3 land in both Gamma and Omicron, the switch point between Beta and Gamma has no effect on the order of fertility. Gamma does not matter to the initial historical order in which techniques are introduced here.

For Omega, the switch point between the Alpha and Gamma techniques does not matter for the order of efficiency. More could be set about the variation in the order of efficiency with the rate of profits.

5.2 The Order of Rentability

The order of rentability varies with intersections of rent curves. These variations in the order of rentability are independent from variations in the order of efficiency. The owner of a less fertile land can obtain more rent per acre than the owner of a more fertile land. As usual, a simplistic marginal productivity theory of distribution cannot be sustained.

5.3 The Cost-Minimizing Technique for the Example

Observations have been made above about the cost-miniziming technique. But the interaction between quantity flows, prices, and rents suggest that the cost-minimizing technique cannot always be found by constructing a frontier from the wage curves.

Figure 5 justifies that Omicron is cost-minimizing at a small rate of profits. Capitalists can gain extra profits by adopting process V in the range of the rate of profits at which both Omicron and Omega pay positive rents. Type 5 land becomes fully farmed by combining the two processes on Type 3 land, and the Omega technique results.

Figure 5: Extra Profits at Omicron Prices

Figure 6 justifies that Tau is cost-minizing at a high rate of profits. Here, too, the Omega technique will be adopted in the range of the rate of profits at which both the technique under consideration and Omega pay positive rents.

Figure 6: Extra Profits at Tau Prices

6.0 Conclusion

This post has illustrated that the introduction of natural resources into a model of the production of commodities provides a complicated picture. The concepts of the order of efficiency and the order of rentability apply to a model in which both intensive and extensive rent occur, depending on net output and the range of the rate of profits. The order of efficiency depends on the sequence in which techniques are cost-minimizing as net output expands, in a more complicated way than when just extensive rents are available. Intensive rent can be transient. An expansion of net output may replace a technique in which intensive rent obtains, at a given rate of profits, with a technique with only extensive rent. And the opposite can happen, as well.

Nonsense In Mankiw's Introductory Textbook

Marginalist economics was shown to be incoherent about two thirds of a century ago. It collapsed just around the issues Marx investigated more than a century and a half ago. How does the ownership of capital goods result in the owner obtaining a return? Mainstream economists address their inadequacy by refusing to talk about their demonstrated inconsistencies.

Those who understand the theory have available a certain form of amusement. They can quickly locate confusion in mainstream textbooks. I happen to have available the eighth edition of N. Gregory Mankiw's Principles of Economics (2018). I may have missed something. Over the course of hundreds of pages, he confuses capital, as a factor of production supplied by households, physical capital goods, deferred consumption, and finance.

Mankiw is careful, I guess, in what he does not say. He has "capital" meaning physical goods, for a while. There seems to be no explanation of the level of interest or dividend payments to households. Households trade consumption between now and later. These savings are not related to changes in the capital stock, although a later section on savings and investment confusingly suggests that some unspecified relationship exists. An aggregate production function has an argument for physical capital, with no discussion of units of measurement. And this all falls by the wayside when he gets to macroeconomics. He presents the obsolete theory of loanable funds, even with silliness about the crowding-out effect of government deficits.

Section 2-1d is "Our first model: the circular flow diagram." With the usual confusion, in one half of the diagram, households supply firms with the factors of production. Capital is "building and machines". At this point, you have a blast furnace in your back yard, which you rent to a steel manufacturer.

Chapter 18 is "The Markets for Factors of Production", and Mankiw emphasizes labor markets. The non-wage part of the national income "went to landowners and to the owners of capital - the economy's stock of equipment and structures - in the form of rent, profit, and interest" (pp. 361-362). Mankiw does not seem to know of any difficulties raised for labor markets or the supposed marginal productivity theory of distribution by the Cambridge capital controversy. "Put simply, highly productive workers are highly paid, and less productive workers are less highly paid" (p. 37).

Capital is like land. "The purchase price of land or capital is the price a person pays to own that factor of production indefinitely. The rental price is the price a person pays to use that factor for a limited period of time" (p. 375). A box on p. 376 is titled "What is capital income?" He brings up interest, dividends, and retained earnings but has no explanation for their levels:

"In our analysis, we have been implicitly assuming that households own the economy’s stock of capital - ladders, drill presses, warehouses, and so on ... In fact, firms usually own the capital they use, and therefore, they receive the earnings from this capital... [I]nstitutional details are interesting and important, but they do not alter our conclusion about the income earned by the owners of capital. Capital is paid according to the value of its marginal product, regardless of whether this income is transmitted to households in the form of interest or dividends or whether it is kept within firms as retained earnings."

Chapter 21 is the theory of consumer choice. Mankiw has the analysis of the trade-off between leisure and work. Section 21-4c treats "How Do Interest Rates Affect Household Saving?" Figure 15 shows the budget constraint and indifference curves for an example of intertemporal choice (p. 444).

Chapter 25 is "Production and Capital" and is part of the treatment of macroeconomics. A box on the production function is on p. 523. Section 25-3a is "Savings and Investment":

"Because capital is a produced factor of production, a society can change the amount of capital it has. If today the economy produces a large quantity of new capital goods, then tomorrow it will have a larger stock of capital and be able to produce more goods and services. Thus, one way to raise future productivity is to invest more current resources in the production of capital. Because resources are scarce, devoting more resources to producing capital requires devoting fewer resources to producing goods and services for current consumption. That is, for society to invest more in capital, it must consume less and save more of its current income. The growth that arises from capital accumulation is not a free lunch: It requires that society sacrifice consumption of goods and services in the present to enjoy higher consumption in the future."

I do not know what skipping my dinner has to do with manufacturing more ladders to outfit employees of firms with orchards and apples to be picked. Neither does Mankiw, of course.

Chapter 26 treats Saving, Investment, and the Financial System. "Now the interest rate is the price that adjusts to balance supply and demand ... for funds in financial markets" (p. 542). Banks and mutual funds are "financial intermediaries" "directing the resources of savers into the hands of borrowers." Mankiw presents the usual national income accounting, with savings and investment in monetary (financial units). "In the language of macroeconomics, investment refers to the purchase of new capital, such as equipment or buildings." He has the crudest loanable funds model. He presents the argument that government deficits crowd out private investment (p. 554) as if it were scientific fact. (On page 590, a box from David Neumark has the usual archaic nonsense about minimum wages causing structural unemployment.)

Mankiw's textbook lacks an explanation of the returns to ownership and an acknowledgement of the existence of this gap. He could argue that this reflects mainstream economics, which is apologetics.

An Example With A Cost-Minimizing Technique With Intensive And Extensive Rent

Figure 1: Detail on Variation of Rent per Acre with Rate of Profits

1.0 Introduction

Consider a model of the production of commodities with non-produced means of production that are unchanged by their use in production. In other words, they are types of land. In a simple model of extensive rent, a single agricultural commodity, 'corn', can be produced, on each type of land, with a single production production. This post expands a simple multi-commodity model to postulate the existence of two production processes on one type of land. The model then combines intensive and extensive rent, depending on the choice of technique.

In the example, all three types of land are at least partially cultivated to satisfy requirements for use. Whether or not all three types of land obtain a rent depends on the level of profits. A mixture of intensive and extensive rent is obtained only for a range of the rate of profits.

I repeat a lot from a previous post so that this post somewhat makes sense by itself.

2.0 Technology, Resources, Final Demand, and Feasibility

A model of the production of commodities is specified by the technology, the endowments of unproduced natural resources, and the requirements for use. Technology is specified, in a discrete technology, by coefficients of production for each production process. Each process is assumed to require the same time to complete and to exhibit constant returns to scale, up to the limited imposed by scarce land. The endowments of each type of land are specifed. Requirements for use are specified by final demand.

Table 1 presents coefficients of production for the example. Two commodities are produced, iron and corn. Aside from the use of land, joint production is not possible. Multiple types of land (that is, three types) exist. Only one agricultural commodity, corn, can be produced on the processes in which land is used. For one type of land, more than one process can be operated on land. Only one process is known for producing iron, the industrial commodity. Each column in Table specifies the person-years of labor, acres of a type of land, tons of iron, and bushels of corn needed to produce a unit output of the specified commodity.

Table 1: The Coefficients of Production

Input	Industry
	Iron	Corn
	I	II	III	IV	V
Labor	1	0.517	91/250	0.67	3/10
Type 1 Land	0	0.49	0	0	0
Type 2 Land	0	0	0.59	0	0
Type 3 Land	0	0	0	9/20	3
Iron	9/20	0.03744	0.0009	0.067	0.08
Corn	2	0.048	0.27	0.15	0.15

Various techniques (Table 2) can be defined with this technology. All twenty-four letters in the Greek alphabet are needed to specify the techniques. Not all techniques are feasible, given technology, endowments, and requirements for use. Land is not scarce for the Alpha, Beta, Gamma, and Delta techniques, and ownership of land obtains no rent. The Epsilon through Upsilon techniques are examples of extensive rent. One type of land obtains a rent in the Epsilon through Xi techniques. All three types are farmed in Omnicro through Upsilon, and two types obtain a rent. Phi is an example of intensive rent. Chi, Psi, and Omega are examples of the combination of intensive and extensive rent.

Table 2: Techniques of Production

Technique	Processes	Land
		Type 1	Type 2	Type 3
Alpha	I, II	Partially farmed	Fallow	Fallow
Beta	I, III	Fallow	Partially farmed	Fallow
Gamma	I, IV	Fallow	Fallow	Partially farmed
Delta	I, V	Fallow	Fallow	Partially farmed
Epsilon	I, II, III	Partially farmed	Fully Farmed	Fallow
Zeta	I, II, IV	Partially farmed	Fallow	Fully Farmed
Eta	I, II, V	Partially farmed	Fallow	Fully Farmed
Theta	I, II, III	Fully Farmed	Partially farmed	Fallow
Iota	I, III, IV	Fallow	Partially farmed	Fully Farmed
Kappa	I, III, V	Fallow	Partially farmed	Fully Farmed
Lambda	I, II, IV	Fully Farmed	Fallow	Partially farmed
Mu	I, III, IV	Fallow	Fully Farmed	Partially farmed
Nu	I, II, V	Fully Farmed	Fallow	Partially farmed
Xi	I, III, V	Fallow	Fully Farmed	Partially farmed
Omnicron	I, II, III, IV	Partially farmed	Fully Farmed	Fully Farmed
Pi	I, II, III, V	Partially farmed	Fully Farmed	Fully Farmed
Rho	I, II, III, IV	Fully Farmed	Partially farmed	Fully Farmed
Sigma	I, II, III, V	Fully Farmed	Partially farmed	Fully Farmed
Tau	I, II, III, IV	Fully Farmed	Fully Farmed	Partially farmed
Upsilon	I, II, III, V	Fully Farmed	Fully Farmed	Partially farmed
Phi	I, IV, V	Fallow	Fallow	Fully Farmed
Chi	I, III, IV, V	Fallow	Fully Farmed	Fully Farmed
Psi	I, II, IV, V	Fully Farmed	Fallow	Fully Farmed
Omega	I, II, III, IV, V	Fully Farmed	Fully Farmed	Fully Farmed

I assume that 100 acres of each of the three types of land are available. Net output consists of 66 tons iron and 88 bushels corn. This completes the specification of the example. The parameters for the example are fairly arbitrary. They are chosen to ensure a reswitching of the order of rentability for the Tau technique and to ensure that the Omega technique is feasible.

Under these assumptions, Omnicron, Rho, Tau, and Omega are feasible. All three types of land are farmed under these three techniques. Type 1 land is only partially farmed under Omnicron, and it is non-scarce and does not obtain a rent. Type 2 land does not obtain a rent under Rho. Type 3 land does not obtain a rent under Tau. All three types are fully farmed under Omega. A linear combination of processesare IV and V are operated side-by-side under Omega. Type 3 land is therefore scarce under Omega. All three types are farmed under Omnicron, with non-scarce Type 3 land only partially farmed.

3.0 Prices of Production

A system of equations specify prices of production for each technique. All operated processes pay the same rate of profits. Rents and wages are paid out of the surplus at the end of the year. A type of land that is only partially farmed is not scarce and pays no rent. I take the net output as the numeraire.

As an example, the system of equations in following five displays specify the prices of production for the Omega technique.

(p₁ a_1,1 + p₂ a_2,1)(1 + r)+ w a_0,1 = p₁

(p₁ a_1,2 + p₂ a_2,2)(1 + r) + rho₁ c_1,2 + w a_0,2 = p₂

(p₁ a_1,3 + p₂ a_2,3)(1 + r) + rho₂ c_2,3 + w a_0,3 = p₂

(p₁ a_1,4 + p₂ a_2,4)(1 + r) + rho₃ c_3,4 + w a_0,4 = p₂

(p₁ a_1,5 + p₂ a_2,5)(1 + r) + rho₃ c_3,5 + w a_0,5 = p₂

Prices of production for the other techniques are specified by a subset of the system of equations for the Omega technique. Each operated process corresponds to an equation in the corresponding system of prices of production. The rent on land that is partially farmed is zero in the corresponding equation, since land in excess supply is not scarce.

The numeraire is specified by a further equation, where the column vector d represents net output.

p₁ d₁ + p₂ d₂ = 1

3.1 On the Solution

A linear combination of the last two equations in the system of prices of production, for the Phi, Chi, Psi, and Omega techniques, eliminates the rent of type 3 land. In the techniques with extensive rent, one of the equations for a corn-producing process does not contain a term for rent either.

This equation for a corn-producing process or the linear combination of the last two equations can be combined with the first equation, for the iron-producing process. This results in a system of two equations in four unknowns, the price of iron, the price of corn, the wage, and the rate of profits. The equation for the numeriare removes one degree of freedom. If the rate of profits is taken as given, this is a linear system which can be solved for prices of produced commodities and the wage.

The rent per acre can be found for each equation remaining in the original system of equations for a technique. The Alpha, Epsilon, Zeta, Eta, Omnicron, and Pi techniques, for example, have the same solution for prices of produced commodities and the wage. Epsilon, Omnicron, Pi have the same rent per acre on type 2 land. Zeta and Omnicron have the same rent per acre on Type 3 land, while Eta and Pi have the same rent per acre on Type 3 land.

3.2 Wage and Rent Curves

Given the technique, the wage is therefore a function of the rate of profits. Likewise the rent on lands that are always fully-farmed with that technique is also a function of the rate of profits.

The wage is a declining function of the rate of profits in the first four techniques and in the 16 techniques with extensive rent alone. A maximum wage corresponds to a rate of profits, and a maximum rate of profits corresponds to a wage of zero. The wage curve can be upward-sloping in models of extensive rent. The wage curves, in the example, happen to be downward-sloping in the example. Figure 2 shows the wage curves for the feasible techniques in the example. The order of efficiency is the order in which techniques are adopted with increasing net output at a given wage or rate of profits. In models with extensive rent, the order of efficiency can be read off the order of wage curves.

Figure 2: Wage Curves for Feasible Techniques

Figure 3 shows the rent curves for the techniques with non-negative rents in the example. Figure 1, at the top of the post, is an enlargement. Rent curves do not need to have any particular slope. They can slope down or up and vary along their extent. The rent curves for Tau are an example of the reswitching of the order of rentability.

Figure 3: Rent Curves for Feasible Techniques

4.0 The Choice of Technique

Only two techniques, Tau and Omega, are feasible in the example and have non-negative rents for scarce lands. Table 3 lists approximate ranges of the rate of profits and which techniques are cost-minimizing in which ranges. The orders of efficiency and the order of rentability are also shown.

Table 3: Cost-Minimizing Technique

Range	Technique	Order of Efficiency	Order of Rentability
0 ≤ r ≤ 29.05 %	Omega	Type 2, 1, 3	Type 1, 2, 3
29.05 ≤ r ≤ 35.50 %	Tau
35.05 ≤ r ≤ 43.76 %			Type 2, 1, 3

Figure 4 justifies which technique is cost-minimizing in which range of the rate of profits. Capitalists can gain extra profits by adopting process V in the range in which Omega pays positive rents. Type 5 land becomes fully farmed by combining the two processes on Type 3 land, and the Omega technique results. For higher rates of profits, Tau is cost-minimizing, up to the maximum for Tau.

Figure 4: Extra Profits at Tau Prices

Intensive and extensive rents are both obtained by landlords when the Omega technique is cost-minimizing. Whenever the Omega technique is cost-minimizing, and in some range of the rate of profits in which Tau is cost-minimizing, the order of efficiency varies from the order of rentability. Type 2 land is more efficienct or more fertile than Type 1 land. Yet ownership of Type 1 land obtains more rent per acre than Type 2 land. Why would one ever expect competitive capitalist markets to reward efficiency?

5.0 Conclusion

This post presents the first concrete example of a case where a cost-minimizing technique combines intensive and extensive rent. It demonstrates that the concepts of the order of effiency and the order of rentability apply to models with intensive rent. As with models with only extensive rent, the order of effiency cannot be generally defined in terms of physical properties alone. And these orders can differ from one another at some given wage or rate of profits.

The example does not illustrate issues that can arise with intensive rent. Wage curves can slope up. The cost-minimizing technique can be non-unique away from switch points. No cost-minimizing technique may exist, even though feasible techniques exist at a given wage or rate of profits (D'Agata 1983).

The analysis can be extended to more kinds of rent and more complicated production models, while still not treating general joint production. Absolute rent, which may not make sense (Basu 2022) and external intensive rent (Kurz and Salvadori 1995) are examples. Rent might be analyzed in models with systematic, persistent variations in the rate of profits among industries (Vienneau 2024). Likewise, a more general model could have some types of lands that are inputs into processes that each produce a different agricultural commodity. Does it make sense to compare and contrast the order of efficiency and the order of rentability in these models?

References

• D’Agata, A. 1983. The existence and unicity of cost-minimizing systems in intensive rent theory, Metroeconomica 35: 147-158.
• Basu, Deepankar. 2022. A reformulated version of Marx's theory of ground rent shows that there cannot be any absolute rent. Review of Radical Political Economics 54(4): .
• Kurz, H. D. and Salvadori N. 1995. Theory of Production: A Long-Period Analysis, Cambridge: Cambridge University Press.
• Quadrio-Curzio, A. 1980. Rent, income distribution, and orders of efficiency and rentability, in Pasinetti, L. L. (ed.) Essays on the Theory of Joint Production, New York: Columbia University Press.
• Quadrio-Curzio, A. and F. Pellizzari. 2010. Rent, Resources, Technologies. Berlin: Springer. [I NEED TO READ THIS TO ENSURE THAT I AM ORIGINAL]
• Vienneau, R. L. 2022. Reswitching in a model of extensive rent. Bulletin of Political Economy 16(2): 133-146.
• Vienneau, R. L. 2024. Characteristics of labor markets varying with perturbations of relative markups. Review of Political Economy (36)2: 827-843.