Why Is Marginalist Economics Wrong?

Because of its treatment of capital. Other answers are possible.

This post draws heavily on the work of Pierangelo Garegnani. I start with a (parochial) definition of economics:

"Economics is the science which studies human behavior as a relationship between ends and scarce means which have alternative uses." -- Lionel Robbins (1932)

The scarce means are the factors of production: land, labor, and capital. Land and labor are in physical terms, in units of acres and person-years, respectively. They can be aggregated or disaggregated, as you wish.

But what is capital? Some early marginalists, such as Knut Wicksell took it as a value quantity, in units of dollars or pounds sterling. Maybe I should rather say, it is given in numeraire units. Capital is taken as given in quantity, but variable in form. The form is a matter of the specific quantities of specific plants, semi-finished goods, and so on.

The goal of the developers of this theory was to explain what Alfred Marshall called normal prices, in long period positions. This theory is inconsistent. As the economy approaches an equilibrium, prices change. The quantity of capital cannot be given a priori. It is both outside and inside the theory.

Leon Walras had a different approach. He took as given the quantities of the specific capital goods. He also included a commodity, perpetual net income, in his model. This is a kind of bond, what households who save may want to buy.

In a normal position, a uniform rate of return is made on all capital goods. Walras also had supply and demand matching. The model of capital formation is overdetermined and inconsistent. Furthermore, not all capital goods may be reproduced in Walras' model. (What did William Jaffe and Donald Walker think of this reading?)

In the 1930s and 1940s, certain marginalists, particularly Erik Lindahl, F. A. Hayek and J. R. Hicks, dropped the concept of a long-period equilibrium. They no longer required a uniform rate of profits in their model. The future is foreseen in their equilibrium paths. If a disequilibrium occurs, no reason exists for the economy to approach the previous path. Expectations and plans are inconsistent. An equilibrium path consistent with the initial data has no claim on our attention.

I am skipping over lots of variations on these themes. I do not even explain why, generally, the interest rate, in equilibrium, is not equal to the marginal product of capital. Or point out any empirical evidence for this result.

A modernized classical political economy, with affinities with Marx, provides a superior approach.

Selected References

• Barnett, Willam A. and Running Han. 2025. Economic Bifurcation and Chaos. World Scientific Publishing. [I have not read this]
• Dorfman, Robert, Paul Samuelson, Robert Solow. 1958. Linear Programming and Economic Analysis.
• Gram, Harvey and G. C. Harcourt. 2020. Keynesian uncertainty: the great divide between Joan Robinson and Paul Samuelson in their correspondence and public exchanges.
• Rosser, J. Barkley. 2013. From Catastrophe to Chaos: A General Theory of Economic Discontinuities, 2nd edition.
• Shell, Karl (ed.) 1967. Essays on the Theory of Optimal Economic Growth. MIT Press.

A Second Double-Fluke Switch Point For A Triple-Switching Example

Figure 1: Extra Profits at Gamma Prices for the Second Double-Fluke Switch Point

This post is a continuation of this series of posts.

A switch point in which wage curves are tangent on the axis for the rate of profits is a double-fluke case symmetrical to the double-fluke case in the previous post. As shown in Figure 1, this case arises in this example as well. The roles of the Alpha and Gamma techniques are reversed. Alpha is always cost-minimizing, while Gamma is cost-minimizing only at the switch point.

This symmetry extends to partitions of the parameter space, as seen in Figure 2. A locus corresponding to a switch point at which wage curves are tangent bounds a region, 1 or 4, in which no switch points exist. Reswitching occurs in regions 3 and 5, which are on the other side of this boundary. This boundary is tangent to a locus corresponding to the fluke property of a switch point being at the extremes of possible rates of profits, zero or its maximum. The point of tangency in the parameter space corresponds to the double-fluke case under examination.

Figure 2: Partitions of the Parameter Space around the Second Double-Fluke Case

Table 1: Cost-Minimizing Technique byRegion

Region	Cost-Minimizing Technique	Notes
1	Alpha	No switch point.
2	Alpha, Gamma	Around the switch point, a lower rate of profits is associated with a LESS round-about technique and greater output per worker.
3	Gamma, Alpha, Gamma	Around the second switch point, a lower rate of profits is associated with a LESS round-about technique and LOWER output per worker.
4	Gamma	No switch point.
5	Alpha, Gamma, Alpha	Around the first switch point, a lower rate of profits is associated with a LESS round-about technique. Around the second switch point, a lower rate of profits is associated with LOWER output per worker.

So far, the partitions in the parameter space have not outlined a region in which triple-switching occurs. Reswitching occurs in regions 3 and 5.

Machinery And The Honesty Of David Ricardo

Consider the introduction of new, advanced machinery into a capitalist economy. This will raise productivity and be good for the population as a whole. It will displace workers, at least temporarily, who were previously making the product of the machine with handicraft production or now obsolete machines with lower productivity. But the production of the machines requires workers too. So, ignoring short-run frictions, will the workers not remain as well off?

David Ricardo believed something like this at one point in his life. But he had come to the opposite conclusion when he revised his Principles for the third edition. And he was forthright in saying so. Some of the displaced workers will be more or less permanently unemployed. By the way, this was not a matter of coming to agree with Malthus on a point about effectual demand.

Ricardo's change of mind was not some abstract academic view. This was a time in England shortly after the Luddites were at their peak. The Luddites had been rioting and destroying new machinery being introduced by industrialists. Ricardo's friend, J. R. McCulloch writes to Ricardo, and he immediately saw the potential of these changes (Ricardo, Works, volume 8, pp. 381-386):

Edinburgh 5 June 1821

My dear Sir

I have to apologise for being so long in returning you my best thanks for the valuable present of the third Edition of your great work - I congratulate you on its success - It is the best proof that can be given of the growing attention now paid to this important science; and it must have a powerful influence in furthering the dissemination of sound principles -

At the same time I must say (and I say it with that regret which I ever must feel in differing widely from one to whom I shall always be proud to look up as to my master) that in my humble opinion the Chapter on Machinery in this Edition is a very material deduction from the value of the work... ...Excess of candour has in this instance occasioned your doing a very serious injury to your favourite science - It was certainly proper that you should have renounced your previous opinions the moment you were satisfied of their fallacy; but this may be done in various ways, and I do not think it was at all necessary for you to make a formal recantation - our object never has been and never can be any other than to endeavour to promote the real interests of the science...

However the manner in which you have published your change of opinion is of comparatively little consequence - It is what I consider the extreme erroneousness of the principles to which you have incautiously lent the sanction of your name that has excited my principal regret - It is impossible to fritter away your argument by fencing it about with conditions - If it is good for any thing at all it is conclusive against all employment of machinery - It is not with greater or less gross or net produce that we have the smallest concern in considering this question; but simply whether does machinery produce commodities cheaper or not? If it does not produce them cheaper it will not be erected, and if it does produce them cheaper its erection must be profitable to every class of persons - The example which you have given does not, as far as I can perceive, by any means warrant a single one of the extraordinary conclusions you have drawn from it - You have not said whether the machine worth £7,500 is to last one, ten, or one hundred years -

...Your argument is to be sure hypothetical; but the hypothesis will be thrown aside, and all those who raise a yell against the extension of machinery, and ascribe to it that misery which is a mere necessary consequence of the oppressiveness of taxation, and of the restraints on commerce will fortify themselves by your authority! If your reasoning and that of Mr. Malthus be well founded, the laws against the Luddites are a disgrace to the Statute book -

Let me beg of you to reconsider this subject - A heresy on a mere doctrinal point is of no moment; but really I could not recommend to any of my friends to bestow the least attention on the study of this science, if I was satisfied that it remained yet to be settled whether the reducing of the price of commodities was advantageous or not - Truly if we are not got this length, our disputes about profits and our other remote conclusions ought to afford infinite amusement to the scoffers - But, I, at least, am not in this quandary - I will take my stand with the Mr. Burke of the American war not with the Mr. Burke of the French revolution - with the Mr. Ricardo of the first not of the third edition - Were there nothing else to allege on the subject I should be perfectly satisfied with what I consider the inherent fallacy involved in all the arguments which have been advanced against machinery...

Were I not aware that in all your speculations you are actuated solely by a desire to contribute to the improvement of the science, I should not have presumed to address to you this hasty and ill-digested letter - But I am satisfied that opinions dictated equally by a regard to the interests of the science, and coming from one who is not the least sincere of your admirers, though they may seem erroneous, will claim and meet with your attentive perusal - I am with the greatest regard and esteem

ever faithfully yours

J. R. McCulloch

Those are extracts from a long letter. I have left out many details of the argument.

Ricardo's friendship with Malthus is another testament to his personality. They continually argued that the other was wrong on political economy. Ricardo would lend out his notes on one of Malthus' books (Works, volume 2) to his friends. He did not try to publish them, for they did not make much sense without the text of Malthus' Principles of Political Economy. Malthus explained to Ricardo that he was mistaken, both in person and through a long interchange of letters. It was Malthus' insistence that even in agriculture, no product and its capital advances consist of the same mixture of commodities that induced Ricardo, as I understand it, to take up the labor theory of value.

Anyways, despite these persistent disagreements, Ricardo continued as a life-long friend of Malthus. I do not think I have that temperament.

Edit: Reference as suggested in coments:

• Paul A. Samuelson. 1989. Ricardo was right! Scandinavian Journal of Economics 91(1): 47-62.

A Double-Fluke Switch Point For A Triple-Switching Example

Figure 1: The Wage Frontier for a Double-Fluke Switch Point

This post is an expansion of a previous one. That post defines the technology and the price system for the three techniques conprising the technology. In Alpha, labor and corn inputs are used to produce corn. Beta and Gamma are more roundabout. In each, Labor and corn are first used to make a machine that lasts two years and can be used to produce corn each year. In Beta, the machine is discarded after being operated one year. The machine is operated for its full life of two years in Gamma. The technology varies over time. One pararemter specifies the decrease in coefficients of production for Alpha.

The solutions of the price system for a technique yields a wage curve. Figure 1 plots the wage curves for the three techniques for a selected time and rates of decrease of the coefficients of production. The cost-minimizing technique at a given rate of profits is the technique with the maximum wage at that rate. In the illustrated example, Gamma is always cost-minimizing, and Alpha is also cost-minimizing at a rate of profits of zero. The switch point between Alpha and Gamma is a fluke in two ways. It is on the wage axis, and the Alpha and Gamma wage curves are tangent at the switch point.

The outer frontier has certain properties in models of pure fixed capital. Wage curves are downward-sloping on the frontier. A maximum wage corresponds to a rate of profits of zero, and a maximum rate of profits corresponds to a wage of zero. The intersection of a wage curve with the wage axis is the output of numeraire per worker for the technique in a stationary state. In a stationary state, the net output is consumed. A higher value of capital per worker goes hand-in-hand with a higher output of corn per worker.

The choice of technique between the Alpha and Gamma techniques, at a given rate of profits, can also be analyzed by examining whether of not extra profits can be obtained in operating the only process comprising the Alpha technique when prices of production for Gamma prevail. The cost of the seed corn and the services of labor can be summed for this process, when operated at unit level. The cost of the seed corn includes a charge for the given rate of profits. Extra profits are the difference between revenues and this sum. Figure 2 illustrates that extra profits are always negative for Alpha at this point in the parameter space, except for the switch point at a zero rate of profits. In the remainder of these posts, this method is used to analyze the choice of technique, since the graph in Figure 2 is more visually compelling than the wage frontier in Figure 1. The Beta technique is never cost-minimizing for the parameters examined here.

Figure 2: Extra Profits for Gamma Prices for the Double-Fluke Switch Point

The double-fluke case examined so far occurs at the point in the parameter space highlighted in Figure 3. Each of the two fluke properties of the double-fluke case correspond to a locus in the parameter space. The boundary between regions 2 and 3 is for cases in which a switch point is on the wage axis. It is hard to distinguish this locus by eye in Figure 3 from the boundary between regions 3 and 4, which is for fluke switch points in which the wage curves for Alpha and Gamma are tangent. A third fluke property, a switch point on the axis for the rate of profits, corresponds to the boundary between regions 1 and 2.

Fluke properties of switch points partition the parameter space into regions. The analysis of the choice of technique is qualitatively invariant in each region. Table 1 summarizes the choice of technique in each region. Techniques are listed in order of an increasing rate of profits. One can check that the variation of the order of cost-minimizing techniques among regions is consistent with the fluke properties of the boundaries between regions.

Figure 3: Structural Dynamics around the Double-Fluke Switch Point

Table 1: Cost-Minimizing Technique byRegion

Region	Cost-Minimizing Technique	Notes
1	Alpha	No switch point.
2	Alpha, Gamma	Around the switch point, a lower rate of profits is associated with a LESS round-about technique and greater output per worker.
3	Gamma, Alpha, Gamma	Around the second switch point, a lower rate of profits is associated with a LESS round-about technique and LOWER output per worker.
4	Gamma	No switch point.

So far, the partitions in the parameter space have not outlined a region in which triple-switching occurs. Region 3 is a region in which reswitching occurs.

A Robinsade For Austrian Capital Theory

I take the following long quote from Bohm-Bawerk.

"The entire sum of originary productive forces at Crusoe’s disposal … is a day’s labor which we shall assume to be a 10-hour workday… Let us assume that the fruit harvest [is] enough to enable our castaway to gather the subsistence minimum in nine hours a day, and enough in 10 hours to furnish him with adequate sustenance for complete health and vigor... Crusoe now has a choice between two lines of conduct. One alternative is ... to consume each day the fruits gathered by a full 10 hours’ work... The other alternative is to restrict himself to the subsistence minimum… In that event – but only in that event – he has a tenth hour open in which he makes hunting equipment for future use... Before there can be any real formation of capital, the productive forces necessary to its production must be saved up at the expense of the enjoyment of the moment.

...The ‘expense of the enjoyment of the moment’ need not always entail downright privation. ...If Crusoe’s labor were somewhat more productive, ...the choice offered might be between ‘adequate supply’ and ‘bounteous fare’. It is not a matter of the absolute magnitude of the minimal claims to enjoyment of the moment, but of their relative magnitude in comparison with ‘income’... The essential point is, that the current endowment of productive forces be not devoted entirely to the enjoyment of the ‘moment’ – the present period – so that a portion of them may be reserved for the service of a future period. Behavior of that kind must unquestionably be called a genuine saving of productive forces.

...It is [productive forces] and not the capital goods themselves which are saved up. We are saving of consumption goods, thereby saving up productive forces and thus can in the end use the latter in order to produce capital goods... To complete the act of forming capital it is of course necessary to complement the negative factor of saving with the positive factor of devoting the thing saved to a productive purpose or, in other words, to endow it with the status of an intermediate product... [O]ur Crusoe has continued throughout one month to consume each day only as much fruit as he could gather in nine hours and has devoted the tenth hour of each day to making hunting equipment. As a result ... he has a bow and some arrows and the possibility of obtaining his subsistence with far greater ease and in much greater abundance than before...

...He must choose another possibility if he is to preserve his capital at its previous level. He must devote at least one hour of his daily allotment of 10 working hours to the rehabilitation of his working equipment and may not spend more than a daily maximum of nine hours on hunting and fruit gathering... In order to preserve capital in status quo ante a certain quantity of the productive forces of the current period must be assigned to the service of the future. And that quantity must be at least equal to the total product of the productive forces of prior periods which is consumed during the current period... Consumption during the current period of the yield of all current and prior productive forces combined, must not exceed the total products that can be derived from the productive forces which accrue afresh in the current period." Bohm-Bawerk (1959: 103-104, emphases in original)

Difficulties arise in this story from applying it to a modern economy and addressing questions of how much. Crusoe’s labor is supposed to represent heterogeneous productive forces. The consumption goods saved and the intermediate capital goods produced stand in a certain ratio, as given by prices. Likewise, the ratio of the consumption goods given up in the current period and those thereby obtained in the future is a kind of price, that is, an interest rate. In a more fully elaborated story, more future-oriented consumers save more, drive the interest rate down, and incentivize managers of firms to adopt more capital-intensive, more roundabout techniques of production. This story cannot be sustained, as is demonstrated, for example, by the triple-switching example in Schefold (1980: p. 170).

References

• Bohm-Bawerk, Eugen von. 1959. Capital and Interest: Volume II: Positive Theory of Capital (Trans. By George D. Huncke). South Holland: Libertarian Press.
• Schefold, Bertram. 1980. Fixed capital as a joint product and the analysis of accumulation with different forms of technical progress. In L. L. Pasinetti, ed., Essays on the Theory of Joint Production, New York, Columbia University Press.

A 1D Diagram For A Triple-Switching Example

Figure 1: Triple Switching with Strucutral Economic Dynamics

1.0 Introduction

I have been using fluke switch points to partition two-dimensional slices of parameter spaces. I know, I think, how reswitching can appear and disappear. But I am confused how more switch points can appear. So this post is a start on exploring a triple-switching example.

I have stumbled upon two examples of triple-switching, so to speak. I have not yet replicated Steedman's claim that triple-switching can arise in his corn-tractor model. But then, in my first explorations I had different types of tractors lasting for the same number of years. So I turn to an example from Schefold, which I have explored previously.

I expect to find numerical examples of phenomena that I have not yet seen. For example, consider a reswitching example in which the first switch point has negative real Wicksell effects, and the second switch point has positive real Wicksell effects. So the first switch point is 'non-perverse' so far. But the first switch point can exibit the reverse substitution of labor. Around the switch point, a higher wage is associated with more employment in the corn industry per (gross) bushel corn produced. This occurs when there is a third switch point between -100 percent and zero.

2.0 Technology

Table 1 presents the structure of coefficients of production for an example. Each column shows the inputs and outputs, in physical quantities, when the process is operated at unit level. All processes exhibit constant returns to scale.

Table 1: Coefficients of Production

Input	Process
	I	II	III	IV
Labor	a0, 1	a0, 2	a0, 3	a0, 4
Corn	a1, 1	a1, 2	a1, 3	a1, 4
New Machines	0	0	1	0
Old Machines	0	0	0	1
Output
Corn	0	1	b1, 3	b1, 4
New Machines	1	0	0	0
Old Machines	0	0	1	0

Three techniques (Table 2) are possible with this technology. Under Alpha, labor and corn are used to produce corn directly. No machines are produced. The Beta and Gamma techniques are more roundabout. First, labor and corn are used to build a machine. Labor works with the machine and inputs of corn to produce more corn. Beta and Gamma differ in whether or not the machine is run for its full physical life of two years. The machine is assumed to be able to be costlessly discarded. Under Beta, the machine is only run for one year.

Table 2: Techniques of Production

Technique	Processes
Alpha	II
Beta	I, III
Gamma	I, III, IV

I make some specific assumptions for the values of coefficients of production:

a_{0, 1} = (3/140) e^{1 - φ t}

a_{0, 2} = e^{(1/2) - σ t}

a_{0, 3} = e^{1 - φ t}

a_{0, 4} = (1/3) e^{1 - φ t}

a_{1, 1} = (31/504) e^{1 - φ t}

a_{1, 2} = (1/2) e^{(1/2) - σ t}

a_{1, 3} = (1/4) e^{1 - φ t}

a_{1, 4} = (2/315) e^{1 - φ t}

b_{1, 3} = 1/2

b_{1, 4} = 1/2

I do not claim that this model of technical change is at all realistic. The idea is to end up with parameters that can be perturbed:

• φ: The rate of decrease of coefficients of production for inputs in processes I, III, and IV.
• σ: The rate of decrease of coefficients of production for inputs in process II.

For given values of φ and σ, productivity improves with time.

3.0 Price Systems

Consider prices of production. I assume that, for the selected technique, the same rate of profits is obtained in all operated processes. Wages are paid out of the surplus at the end of the year. Corn is the numeraire.

Figure 1, at the top of the post, shows the cost-minimizing technique at each level of the wage, for the indicated values of φ and σ. The cost-minimizing technique varies with the illustrated structural economic dynamics. Schefold's example arises at t = 10. The wage frontier is not particularly striking. The difference in the wage curves for Alpha and Gamma are barely distinguishable to the eye.

4.0 Price Conclusion

My next step is to graph (σ t) against (φ t) for various fluke cases. Maybe I will present intermediate results before I figure out what the full parameter space looks like.

So now I have several problems queued up:

1. Explore how triple-switching can arise with partitions of a two-dimensional parameter space.
2. See if I can get publsihed my construction of Hayekian triangles from models of prices of production.
3. Update and see if I can get published my working paper with one-dimesional diagrams.
4. Write up another refutation of Austrian claims. Recall I want to mention that this is only part of a larger demonstration that capital-intensity is not to be explicated in terms of a period of production.
5. Write up an exposition of local perturbations of fluke switch points with two-dimensional diagrams.
6. See if I can make sense of the order of efficiency and the order of rentability in a model that combines intensive and extensive rent.

An Expanded Parameter Space For The Reverse Substitution Of Labor

Figure 1: A Larger Parameter Space

This post is an expansion on the first example here. It presents shortly a more comprehensive analysis of the variation in the choice of technique in the example of circulating capital in Section 2. Local perturbations of two coefficients of production are examined there. Figure 1 partitions a larger part of the space defined by these two coefficients of production. Table 1 exhibits how the cost-minimizing technique varies with the rate of profits in each region.

Table 3: Ranges of the Rate of Profits by Region

Region	Range	Technique	Notes
1	0 ≤ r ≤ r1	Beta	Reverse substitution of labor at switch point.
	r1 ≤ r ≤ rα,max	Alpha
2	0 ≤ r ≤ r1	Beta	Switch point is 'non-perverse'.
	r1 ≤ r ≤ rα,max	Alpha
3	0 ≤ r ≤ rβ,max	Beta	No switch point.
4	0 ≤ r ≤ rβ,max	Beta	No switch point.
5	0 ≤ r ≤ r1	Alpha	Switch point is 'non-perverse'.
	r1 ≤ r ≤ rβ,max	Beta
6	0 ≤ r ≤ r1	Alpha	Reswitching. Second switch point exhibits capital-reverseing and the reverse substitution of labor.
	r1 ≤ r ≤ r2	Beta
	r1 ≤ r ≤ rα,max	Alpha
7	0 ≤ r ≤ rα,max	Alpha	No switch point.

Section 2 in the previous focuses on regions 1, 2, 3, and 4. Region 3 and 4 differ in that in region 4, the wage curves intersect at a negative rate of profits greater than -100 percent. This post presents an analysis, in a model of fixed capital, much like the fluke switch point associated with the intersections of the partitions between regions 1, 6, and 7. Does checking how the variation of the analysis of the cost-minimizing technique among regions, summarized in Table 1, relates to the fluke cases defining the partitions in Figure 1 clarify that variation? Perturbations of coefficients of production, in this example of circulating capital, illustrate how reswitching can emerge, as well as the emergence of the reverse substitution of labor.

Did Marginalism Become Accepted As A Reaction to Marxism?

I take it for granted that marginalism became accepted partly because Marx had used the best in classical political economy in his account of why socialism would and should transcend capitalism. This post presents some who have argued for or asserted the same.

I start by summarizing an argument from Antonia Campus. Campus argues that the marginalists in the 1870s did not have an accepted theory of production, cost, and price. Only in the 1890s did the marginal productivity theory of distribution become accepted. Now that that theory has been demolished, in the 1960s, we are back into the confusion of the 1870s, with every person his own capital theorist.

"With the publication in 1867 of Volume 1 of Capital, Ricardo's theory of distribution and value had in fact reappeared, not in the conciliatory form of J. S. Mill's Principles, but in the dangerous one which had been typical of this theory in the decade following Ricardo's death. According to Böhm-Bawerk, this theory constituted for the Germany of 1884 'the focal point about which attack and defence rally in the war in which the issue is the system under which human society shall be organized'.

On account of the impasse in which the theory of distribution was, and the ensuing chaos in economic theory, there was the danger that Ricardo's theory of distribution - in the most advanced elaboration it had found in Volume I of Marx's Capital - might fill the gap, and become even in Britain the 'focal point' in the struggle for and against the established social order. This danger must have seemed not too abstract, in the climate of Socialist revival of the 1880s, and especially after the foundation of the Social Democratic Foundation in 1881 and the Fabian Society in 1883." -- Antonia Campus. 1987. Notes on cost and price: Malthus and the marginal theory. Political Economy: Studies in the Surplus Approach 3(1): 11.

Here is Luigi Pasinetti saying something along the same lines:

"What turned out to be so devastating was the social impact of [Marx's] writings. The immediate practical effect of Marx's call for a social revolution was to elicit a strong social reaction. The establishment of the Western nations, at the end of the nineteenth century, became scared by Marx's revolutionary call. This by itself explains a lot of the fortune that in academic circles blessed marginalism in the 1870s, whose success was essentially analytical...

...In academic circles, this no doubt represented a radical change, but not in the strict sense of a scientific 'revolution', though some historians of economic thought later hastened to call it so (the 'Marginal Revolution'). Conceptually, it was a 'counter-revolution', an anachronistic achievement, yet a beautiful one, reached with the most sophisticated tools of economic analysis (precisely what the Classical economists had lacked).

At the end of the nineteenth and the beginning of the twentieth century, marginal economic theory led to conclusions which were pleasing to the establishment, especially in terms of a splendid detachment from the hot social issues that were boiling up in the real world, and in terms of arguments that could easily be used for the advocacy of unrestricted laissez-faire policies, supposedly leading, in ideal conditions, to optimal positions..." -- Luigi L. Pasinetti (2007).

Pierangelo Garegnani summarizes some of Sraffa's unpublished notes:

"For the school of 'cost', Sraffa is here referring to Ricardo's 'cost value', influenced, Sraffa says, by the author's 'anti-landlord complex' whereby 'rent not entering cost is disgraced'. The second school, that of 'utility', and with it the conflict between the two, came instead into being, Sraffa continues, when Ricardo’s 'cost' theory was 'taken up by Marx and used as a weapon for the workers'. That provoked by reaction the 'immediate simultaneous success' of the utility-based theory of value of Jevons, Menger and Walras – a theory that, significantly enough was ignored when it made its first appearance in the work of authors such as Dupuit and Gossen, before Marx's work created the need to develop a substitute for labor values." – Pierangelo Garegnani. 2005. On a turning point in Sraffa’s theoretical and interpretative position in the late 1920s. European Journal of the History of Economic Thought 12(3).

Gunnar Myrdal says something similar:

"One point which emerges from our analysis of the classical exchange value and real value theory is that Marx's theory of surplus value is not the result of a 'gross misunderstanding.'...Marx was right in saying that his surplus value theory follows from the classical theory of real value...Moreover, Marx was not the first to draw radical conclusions from it. All pre-Marxist British socialists derived their arguments from Adam Smith and later from Ricardo. Economists did not welcome these inevitable conclusions...He thus touched upon a sore point of economic theory and, probably for this reason, caused so much irritation amongst economists. They often tried not so much to prove him wrong, which would not have been too difficult, as to show that he was an utter fool, a bungler, misguided by those despised German philosophers...The classical theory of value leads inevitably to a rationalist radicalism, if not necessarily in Marx's formulation, at any rate in that direction. For the historian of thought the real puzzle is why the classics did not draw these radical conclusions." -- Gunnar Myrdal, The Political Element in the Development of Economic Theory.

I know of W. J. Ashley from Sraffa's unpublished notes:

The marginal conception of value which this generation owes to Jevons and Menger was clearly enough expounded by Longfield in 1833, but it passed unregarded... It is evident that their inattention was due, not to dissatisfaction with what men like Longfield offered them, but to satisfaction with the apparently sufficient formulae they had already mastered...

...Meanwhile ... the dissemination of the teachings of the so-called 'scientific' socialists - of Lassalle's 'Iron Law of Wages,' and Marx's 'Surplus Value' - disposed conservatively minded thinkers to re-examine that Ricardian teaching to which the Socialists, with so much show of reason, were in the habit of appealing." -- W. J. Ashley (1907).

The idea that marginalism became accepted partly as a reaction to Marxism is an established take from those who have examined the question over more than a century. You do not even need to be a radical to believe it.

Other takes emphasize a reaction to Henry George, Mirowski's account of physics envy, and the extension of Ricardo's theory of rent to all so-called factors of production.

Ian Steedman edited Socialism and Marginalism in Economics: 1870-1930 in 1995. The essays in this book are intelligent, informed, and much more nuanced than this post. Some socialists adopted marginalism. George Bernard Shaw was convinced by Philip Wicksteed,\ and remained a Fabian. Apparently, in Denmark it was not even a controversy. Wicksell was a radical in Sweden, advocating birth control. But on economics he wasn't very socialist. Vladimir Dmitriev and Ladislaus Bortkiewicz interpreted Ricardo and Marx with linear production models, a tradition continued by Robert Remak, John Von Neumann, and Wassily Leontief.

Employment And Wages Not Determined By The Supply And Demand Of Labor

Figure 1: The Demand for Labor

1.0 Introduction

Wages and employment are not determined in competitive markets by the interaction of well-behaved supply and demand curves, as portrayed in much introductory economics. At least, no reason exists to thinks so. Every once in a while I like to recall that this is an implication of the Cambridge Capital Controversy.

2.0 Technology

Consider a very simple competitive capitalist economy in which corn and iron are produced from inputs of labor, iron, and corn. All production processes in this example require a year to complete. The managers of firms know of two processes for producing corn and two processes for producing iron (Table 1). The processes a and b, for producing corn, require the tabulated inputs to be available at the beginning of the year for each bushel corn produced and available at the end of the year. Similarly, process c, for example, requires one person-year, 1/40 bushels corn, and 1/10 tons iron to be available at the beginning of the year for each ton of iron produced by this process. This is an example of circulating capital; all inputs of corn and iron are used up during the year in producing the gross output.

Table 1: Coefficients of Production

Input	Industry
	Corn		Iron
	a	b	c	d
Labor	a0, 1(a) = 1	a0, 1(b) = 1	a0, 2(c) = 1	a0, 2(d) = 275/464
Corn	a1, 1(a) = 2/5	a1, 1(b) = 3/5	a1, 2(c) = 1/40	a1, 2(d) = 0
Iron	a2, 1(a) = 2	a2, 1(b) = 1/2	a2, 2(c) = 1/10	a2, 2(d) = 113/232

A technique consists of a process for producing corn and a process for producing iron. Thus, there are four techniques in this example. They are defined in Table 3.

Table 2: Techniques of Production

Technique	Corn Process	Iron Process
Alpha	a	c
Beta	a	d
Gamma	b	c
Delta	b	d

3.0 Quantity Flows

Suppose the net output of the economy is some multiple c of the numeraire. I let d₁ be the bushels corn in the numeraire, and d₂ be the tons iron. How can one find, for a given technique, how much labor must be employed throughout the economy to produce, say, a bushel corn?

Let the coefficients of production be as above for a given technique. Let d₁ be unity, and d₂ be zero. The bushels corn y₁ and tons iron y₂ in net output are now specified. The question becomes what are the gross quantities q₁ of corn and q₂ of iron that need to be produced for the given net output.

With this specification, the following equation must be satisfied for the production of the given net output of corn:

y₁ = c d₁ = q₁ - (a_{1, 1} q₁ + a_{1, 2} q₂)

The following equation is for the given net output of iron:

y₂ = c d₂ = q₂ - (a_{2, 1} q₁ + a_{2, 2} q₂)

The labor L employed throughout the economy with this net output is:

L = a_{0, 1} q₁ + a_{0, 2} q₂

One can set L to unity and solve the above system of equations for c, q₁, and q₂. The person-years of labor needed to produce a net output of one unit of the numeraire is then the reciprocal of c.

Or one can set c to unity and solve for L, q₁, and q₂. This, too, will find the person-years employed throughout the economy, with the given technique, to produce one unit of the numeraire net.

4.0 Prices of Production

Which technique will the firm adopt, if any? The answer depends, in this analysis, on which is more profitable. So one has to consider prices. I here assume that inputs of iron and corn are charged at the start of the year. The wages for labor are paid out of the surplus at the end of the year.

Select a technique.

• p₁: The price of a bushel corn.
• p₂: The price of a ton iron.
• w: The wage for hiring a person-year of labor.
• r: The rate of profits

The corn-producing process gives one equation for specifying prices of production:

(p₁ a_1,1 + p₂ a_2,1)(1 + r) + w a_{0, 1} = p₁

The iron-producing process specifies another equation:

(p₁ a_1,2 + p₂ a_2,2)(1 + r) + w a_{0, 2} = p₂

The price of the numeraire is unity:

p₁ d₁ + p₂ d₂ = 1

You can solve the above system, to find the price of corn, the price of iron, and the wage as (rational) functions of the rate of profits. You actually want to solve for functions of (1 + r). Each technique yields a set of functions.

5.0 The Choice of Technique

Figure 2 graphs the wage curves for each of the four techniques. The outer frontier shows the cost-minimizing technique. The sequence of cost-minimizing techniques, as the wage increases, is Beta, Alpha, Gamma, and Delta. Apparently, I constructed this example, decades ago, to have all four techniques on the frontier.

Figure 2: Wage Curves and the Their Outer Frontier

The construction of the outer frontier in the analysis of the cost-minimizing technique is a derived result. Figure 3 shows the extra profits, for any wage, to be obtained when Alpha or Beta prices prevail. You can see Beta is cost-minimizing at the smallest range of wages and Alpha at the next largest range. Suppose Alpha was in operation at the lowest wages. Prices would signal to managers of firms that they should make iron with process d, not process c. So they would adopt the Beta technique. Market prices would no longer match prices of production. A disequilibrium process would presumably end up with Beta prices prevailing at the given wage.

Figure 3: Extra Profits at Alpha and Beta Prices

In a model of circulating capital, it does not matter at which system of prices you start at. Away from switch points, only one technique is cost-minimizing. This uniqueness does not necessarily hold for general models of joint production.

By the way, this is also an instance of process recurrence, as well as of capital-reversing. Beta is cost-minimizing at the lowest wage, and Delta at the highest wage. Both operate the second process in iron production. This process recurs. Process recurrence can arise with neither capital reversing nor the reswitching of techniques. I suppose this independence is more apparent if more that two goods are being produced.

6.0 The 'Demand' for Labor

The above has briefly outlined how to find the cost-minimizing technique for any given wage, up to a maximum. And I have also outlined how to find employment for any given technique and net output. Figure 1, at the top of this post, puts these results together to present a graph for the long-period demand for labor. It is not downward-sloping throughout. The existence of capital-reversing implies that labor-demand curves can slope up.

7.0 Conclusion

I really do not know how to explain what I understand most economists teach their students. I suppose some students might greet a upward-sloping labor demand curve by talking about how some make mistakes and it takes time for managers of firms to learn. They might bring up an evolutionary process. Or principal agent problems, transactions costs, and information asymetries. And on and on.

But these imperfections and frictions are off-point. The basic logic of the textbooks is wrong. And this has been known for over a half-century.

Elsewhere

• A 2019 profile of Stephen Marglin in the Harvard Crimson.
• William Morris, the author of News from Nowhere, wrote the Manifesto of the Socialist League.
• David O'Connell explains, in Jacobin, that Catholic orthodoxy is quite radical on the economy.
• Jack London's How I became a socialist.

Local Perturbations Of A Fluke Switch Point For Intensive Rent

Figure 1: A Parameter Space

1.0 Introduction

This is a re-creation and elaboration of a previous post.

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

2.0 Technology

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

Table 1: Coefficients of Production

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

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

Table 2: Techniques of Production

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

3.0 A Fluke Switch Point

Figure 2 shows the wage and rent curves for feasible techniques at a selected parametrization. I take the wage curve for a technique to be defined only for non-negative rates of profits at which the wage, rent per acre, and the prices of produced commodities are non-negative. The wage is negative for Delta for rates of profits below that at the switch point, and rent is negative for rates of profits greater than that at the switch point. Rent is negative for Epsilon for rates of profits less than at the switch point, and the wage is negative for greater rates of profits. Thus, the switch point is the only point on the wage curves for Delta and Epsilon. The switch point is a fluke in at least two ways. It is a switch point for three techniques, not two. And it is on the axis for the rate of profits.

Figure 2: Wage and Rent Curves for a Fluke Case

4.0 Local Perturbations within a Parameter Sapce

Figure 1, at the top of this post, shows a partition of the parameter space around this fluke case. An intersection of three wage curves over the axis for the rate of profits is a combination of three pairs of wage curves intersecting over the axis for the rate of profits. These three fluke cases are the partitions between regions 1 and 2, regions 2 and 3, regions 6 and 7, regions 7 and 8, and regions 8 and 1. The partition between regions 3 and 4 is associated with the fluke case of three wage curves intersecting at a non-negative rate of profits. The partitions between regions 4 and 5 and between regions 5 and 6 illustrate a fluke switch point specific to models of rent.

Regions 2 through 8 illustrate the possible non-uniqueness and non-existence of a cost-minimizing technique. For concreteness, consider the point in region 5 with the wage curves and variation in rent per acre illustrated in Figure 3. For rates of profits up to the first switch point, Alpha is cost-minimizing. Epsilon is cost-minimizing between the switch points, and Delta is also cost-minimizing for high rates of profits in this range. Beyond the second switch point, no technique is cost-minimizing. Whether or not land is scarce depends on the distribution of income.

4.1 The Choice of Technique in Region 5

Figure 3: Wage and Rent Curves in Region 5

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

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

Figure 4: Extra Profits for Delta and Epsilon in Region 5

4.2 Fluke Cases Bordering Region 5

Presenting two fluke switch points might assist in understanding how the analysis of the choice of technique varies in the part of the parameter space examined here. Figure 5 shows the wage and rent curves for a fluke switch point for parameters on the partition between regions 4 and 5. This fluke switch point is associated with the disapperance of the range of the rate of profits, in region 4, where both Alpha and Epsilon are cost-minimizing. It is associated with the emergence, in region 5, of a range of the rate of profits where only Epsilon is cost-minimizing.

Figure 5: A Fluke Switch Point on the Upper Boundary of Region 5

Figure 6, on the other hand, shows a fluke switch point for parameters on the boundary of regions 5 and 6. This switch point is associated with the disappearance of a range of the rate of profits where only Alpha and Epsilon have defined wage and rent curves and neither technique is cost-minimizing. And it is associated with the appearance of a range of the rate of profits where only Alpha and Delta have defined wage and rent curves and neither technique is cost-minimizing. The fluke switch points in Figure 5 and Figure 6 can only arise in models of joint production, including models of rent.

Figure 6: A Fluke Switch Point on the Lower Boundary of Region 5

4.3 Overview of Regions as a Whole

Qualitative properties of the analysis of the choice of technique do not vary within each numbered region. Table 3 describes the variation in the cost-minimizing technique with the rate of profits in each numbered region in Figure 1. In region 1, Alpha is cost-minimizing for all feasible rates of profits. Land is not scarce, and obtains no rent. For a high enough rate of profits in region 2, Alpha and Delta are both non-uniquely cost-minimizing. The wage curve for Delta slopes up and rent per acre decreases with the rate of profits when Delta is operated. The switch point for Alpha and Delta is at a positive wage. For any rate of profits greater than the rate of profits at the switch point, no cost-minimizing technique exists. In region 3, a switch point between Alpha and Epsilon occurs at a rate of profits higher than the maximum rate of profits for Delta. Epsilon is never cost-minimizing.

Table 3: Ranges of the Rate of Profits by Region

Region	Range	Technique	Comment
1	0 ≤ r ≤ rα,max	Alpha	Delta and Epsilon have negative wage or rent throughout.
2	0 ≤ r ≤ rδ,min	Alpha	Alpha has a positive wage.
	rδ,min ≤ r ≤ rδ,max	Alpha & Delta	Alpha has a positive wage; Delta has a positive wage and rent.
	rδ,max ≤ r ≤ rα,max	None	Alpha has a positive wage.
3	0 ≤ r ≤ rδ,min	Alpha	Alpha has a positive wage.
	rδ,min ≤ r ≤ rδ,max	Alpha & Delta	Alpha has a positive wage; Delta has a positive wage and rent.
	rδ,max ≤ r ≤ rε,min	None	Alpha has a positive wage.
	rε,min ≤ r ≤ rε,max	None	Alpha has a positive wage; Epsilon has a positive wage and rent.
	rε,max ≤ r ≤ rα,max	None	Alpha has a positive wage.
4	0 ≤ r ≤ rδ,min	Alpha	Alpha has a positive wage.
	rδ,min ≤ r ≤ rε,min	Alpha & Delta	Alpha has a positive wage; Delta has a positive wage and rent.
	rε,min ≤ r ≤ r1	Delta & Epsilon	Alpha has a positive wage; Delta and Epsilon have a positive wage and rent.
	r1 ≤ r ≤ rδ,max	None	Alpha has a positive wage; Delta and Epsilon have a positive wage and rent.
	rδ,max ≤ r ≤ rε,max	None	Alpha has a positive wage; Epsilon has a positive wage and rent.
	rε,max ≤ r ≤ rα,max	None	Alpha has a positive wage.
5	0 ≤ r ≤ rε,min	Alpha	Alpha has a positive wage.
	rε,min ≤ r ≤ rδ,min	Epsilon	Alpha has a positive wage; Epsilon has a positive wage and rent.
	rδ,min ≤ r ≤ r1	Delta & Epsilon	Alpha has a positive wage; Delta and Epsilon have a positive wage and rent.
	r1 ≤ r ≤ rδ,max	None	Alpha has a positive wage; Delta and Epsilon have a positive wage and rent.
	rδ,max ≤ r ≤ rε,max	None	Alpha has a positive wage; Epsilon has a positive wage and rent.
	rε,max ≤ r ≤ rα,max	None	Alpha has a positive wage.
6	0 ≤ r ≤ rε,min	Alpha	Alpha has a positive wage.
	rε,min ≤ r ≤ rδ,min	Epsilon	Alpha has a positive wage; Epsilon has a positive wage and rent.
	rδ,min ≤ r ≤ r1	Delta & Epsilon	Alpha has a positive wage; Delta and Epsilon have a positive wage and rent.
	r1 ≤ r ≤ rε,max	None	Alpha has a positive wage; Delta and Epsilon have a positive wage and rent.
	rε,max ≤ r ≤ rδ,max	None	Alpha has a positive wage; Delta has a positive wage and rent.
	rδ,max ≤ r ≤ rα,max	None	Alpha has a positive wage.
7	0 ≤ r ≤ rε,min	Alpha	Alpha has a positive wage.
	rε,min ≤ r ≤ rε,max	Epsilon	Alpha has a positive wage; Epsilon has a positive wage and rent.
	rε,max ≤ r ≤ rδ,min	None	Alpha has a positive wage.
	rδ,min ≤ r ≤ rδ,max	None	Alpha has a positive wage; Delta has a positive wage and rent.
	rδ,max ≤ r ≤ rα,max	None	Alpha has a positive wage.
8	0 ≤ r ≤ rε,min	Alpha	Alpha has a positive wage.
	rε,min ≤ r ≤ rε,max	Epsilon	Alpha has a positive wage; Epsilon has a positive wage and rent.
	rε,max ≤ r ≤ rα,max	None	Alpha has a positive wage.

In region 4, a switch point exists on the wage frontier between Alpha and Epsilon, at a rate of profits greater than the minimum rate of profits for Delta. A range of the rate of profits remains at which Alpha and Delta are both non-uniquely cost-minimizing. Above the rate of profits at this switch point, the wage frontier resembles the wage frontier in Figure 3 at rates of profits greater than the minimum rate of profits for Delta. In region 5, the range of the rate of profits at which both Alpha and Delta are cost-minimizing has disappeared. In region 6, the range of the rate of profits has disappeared in which no technique is cost minimizing, but Epsilon has a positive rate of profits and rent and Delta does not.

In region 7, Delta is no longer cost-minimizing at any feasible rate of profits. Alpha is cost-minimizing at a low rate of profits, and Epsilon is uniquely cost-minimizing at any feasible rate of profits greater than the rate of profits at the switch point between Alpha and Epsilon. In region 8, Delta is not only no longer ever cost-minimizing, but Delta never has both a positive rate of profits and rent.

Suppose one is not interested in qualitative variations in ranges of the rate of profits in which no cost-minimizing technique exists. In one range for some regions, Alpha has a positive rate of profits, and Delta and Epsilon each have positive rates of profits and a positive rent. Yet when prices for Delta prevail, extra profits can be obtained by operating processes in Epsilon. And when prices for Epsilon prevail, extra profits come from adopting Delta. In another range of rates of profits, neither Delta nor Epsilon obtain both non-negative rates of profits and rents. Yet Alpha is not cost-minimizing. Ignoring these variations, regions 2 and 3 can be combined. Regions 5 and 6 can be combined. Likewise, regions 7 and 8 can be combined

5.0 Conclusion

Whether or not land obtains a rent can depend on the distribution of income. For a low-enough rate of profits in regions 2 through 8, the first three processes are operated. Iron, steel, and corn are each produced with one process, and land obtains no rent. For a higher rate of profits, the Delta or Epsilon technique can be cost-minimizing. Corn is produced by two processes, and scarce land obtains a rent. Even if the requirements for use can feasibly be satisfied with some land not farmed, the cost-minimizing technique may be such that two processes are operated side-by-side on land, with no land lying fallow. The example illustrates that an examination of fluke switch points can help in understanding qualitative variations in the analysis of the choice of technique, even in a case where certain properties of models of circulating capital do not hold.

References

• D'Agata, Antonio. 1983. The existence and unicity of cost-minimizing systems in intensive rent theory. Metroeconomica, 35: 147-158.
• Kurz, Heinz and Neri Salvadori. 1995. Theory of Production: A Long-Period Analysis. Cambridge: Cambridge University Press.

Some Others Long Ago On Fluke Switch Points

I have been using fluke switch points to partition parameter spaces into regions. In each region, the analysis of the choice of technique does not qualitatively vary.

Fluke switch points have been discussed in the literature on the analysis of the choice of technique. Mostly, these mentions dismissal fluke cases, on the correct grounds that they only occur at an accidental point in the parameter space.

Here is a statement from one contribution to the famous QJE symposium:

"Cases with multiple roots or cases in which the curves cross only at end points … can be classified as irrelevant since the F[actor] P[rice] F[rontier] (envelope) is unchanged by their exclusion." -- Bruno, Michael, Edwin Burmeister, and Eytan Sheshinski. 1966. The nature and implications of the reswitching of techniques. Quarterly Journal of Economics, 80 (4): 534.

And another statement from the same contribution:

"'Adjacent' techniques on two sides of a switching point will usually differ from each other only with respect to one activity." -- Bruno, Michael, Edwin Burmeister, and Eytan Sheshinski. 1966. The nature and implications of the reswitching of techniques. Quarterly Journal of Economics, 80 (4): 542.

Here is from Garegnani's contribution:

"The possibility that, at r* and r**, the two curves touch without intersecting is excluded…" -- Garegnani, Pierangelo. 1966. Switching of techniques. Quarterly Journal of Economics, 80 (4): 567

I suppose I can look for more.

Three Examples For The Cambridge Capital Controversy

Figure 1: A Parameter Space

1.0 Introduction

I have been reconstructing some of my examples. The first example in this post is from here. I am thinking of writing a draft article, as mentioned here. While I am at it, I thought I would also work through the examples in Garegnani (1966) and Bruno, Burmeister & Sheshinski (1966), both from the symposium in the Quarterly Journal of Economics of that year.

2.0 The Emergence of the Reverse Substitution of Labor

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

Table 1: Technology for the Reverse Substitution of Labor

Input	Industry
	Iron	Corn
		Alpha	Beta
Labor	a0,1=1	aα0,2=16/25	aβ0,2
Iron	a1,1=9/20	aα1,2=1/625	aβ1,2
Corn	a2,1=2	aα2,2=12/25	aβ2,2=27/400

Two parameters are not given numerical values in this specification of technology. The approach taken here is to examine a local perturbation of parameters in a two-dimensional slice of the higher dimensional parameter space defined by the coefficients of production in particular numeric examples. With wages paid out of the surplus product at the end of the period of production, the wage curves for the two techniques are depicted in Figure 2 for a particular parametrization of the coefficients of production. The Beta technique is cost-minimizing for any feasible distribution of income. If the wage is zero and the workers live on air, the Alpha technique is also cost-minimizing.

Figure 2: Wage Curves with Two Fluke Switch Point

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

Figure 1 depicts a partition of the parameter space around the point with these two fluke switch points. Below the horizontal line, the switch point on the axis for the rate of profits has disappeared below the axis. The Beta technique is cost-minimizing for all feasible non-negative rates of profits. Above this locus, the Alpha technique is cost-minimizing for a low enough wage or a high enough feasible rate of profits.

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

In the northeast of Figure 1, the switch point for a positive rate of profits exhibits the reverse substitution of labor. Around this switch point, a higher wage is associated with the adoption of a process producing the consumer good in which more labor is employed per unit of gross output. The other switch point exists for a rate of profits between -100 percent and zero. Steedman (2006) presents examples with this phenomenon in models with other structures

Qualitative changes in the wage frontier exist in the parameter space away from the part graphed in Figure 1. The analysis presented here is of local perturbations of the depicted fluke case.

2.0 Example from Garegnani (1966)

I think of Luigi Pasinetti as the first to show that David Levhari's non-(re)switching theorem is false. But the counter-example that he presented at the September 1965 Rome Congress of the Econometric Society did not quite meet all of the assumptions of Levhari's theorem.

Table 2 defines the coefficients of production for the counter-example from Pierangelo Garegnani's paper in the QJE symposium devoted to the topic. Figure 3 presents the wage curves for the example. Switch points are at 10 percent and 20 percent, appealingly reasonably small rates of profits. But the wage curves are visually hard to distinguish. The switch points are more apparent in the plot of extra profits at Alpha prices, in the right pane.

Table 2: Technology for a Reswitching Example

Input	Industry
	Iron	Corn
		Alpha	Beta
Labor	a0,1=89/10	aα0,2=9/50	aβ0,2=3/2
Iron	a1,1=0	aα1,2=1/2	aβ1,2=1/4
Corn	a2,1=379/423	aα2,2=1/10	aβ2,2=5/12

Figure 3: Wage Curves for a Reswitching Example

In some sense, it is unfair to criticize scholars of that time for not creating more apparent examples. The tools I have are much more advanced for seeing the effect of perturbing a coefficient. And, nevertheless, I still have some examples that are hard to see the 'perverse' results.

3.0 Example from Bruno, Burmeister & Sheshinski (1966)

The counter example from Michael Bruno, Edwin Burmeister, and Eytan Sheshinski's paper in the QJE symposium has more a visually striking wage frontier. Table 3 presents the coefficients of production. (I have reordered the industries.) Figure 4 plots the wage curves. The switch points are at approximately 46.58 percent and 166.88 percent or wages of approximately 0.8065 and 0.2595 bushels per person-year.

Table 3: Technology for Another Reswitching Example

Input	Industry
	Iron	Corn
		Alpha	Beta
Labor	a0,1=1	aα0,2=33/100	aβ0,2=1/100
Iron	a1,1=0	aα1,2=1/50	aβ1,2=71/100
Corn	a2,1=1/10	aα2,2=3/10	aβ2,2=0

Figure 4: Wage Curves for another Reswitching Example

Many like to quote Paul Samuelson declaration that:

"...the simple tale told by Jevons, Böhm-Bawerk, Wicksell, and other neoclassical writers - alleging that, as interest rate falls in consequence of abstention from present consumption in favor of future, technology must become in some sense more 'roundabout,' more 'mechanized,' and more 'productive' - cannot be universally valid." -- Paul A. Samuelson (1966).

Bruno, Burmeister & Sheshinski are just as clear:

"Numerical examples and the realization that switching points are roots of n-th degree polynomials (and therefore numerous) have convinced us that reswitching may well occur in a general capital model." - Bruno, Burmeister & Sheshinski (1966, p. 527)

Somehow, empirical work has not made it apparent all of these possible real roots, despite the exploration of economies with many industries. I like this quotation too:

"Although the latter sufficiency condition is again highly restrictive, it may be somewhat less restrictive than the former one: note the latter allows changes of single activities while the former does not. We might also observe that the latter condition seems to be the most natural extension of our previous two-sector nonswitching theorem... Let us again stress that, except for highly exceptional circumstances, techniques cannot be ranked in order of capital intensity. We thus conclude that reswitching is, at least theoretically; a perfectly acceptable case in the discrete capital model." - Bruno, Burmeister & Sheshinski (1966, p. 545)

I skimmed the sufficiency condition. I think technologies with different capital goods used in different techniques are ruled out. Likewise, processes in the same industry in which some capital goods are increased and others are decreased might also be ruled out. It is the general case that technology can be such that reswitching is possible.