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

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

This post is a continuation of this series of posts.

In the last double-fluke case, the three switch points between Alpha and Gamma coincide as a ingle switch point. Figure 1 illustrates, while Figure 2 depicts how the parameter space is partitioned around this double-fluke case. Region 7, in which one switch point occurs, is connected. At the point corresponding to the double-fluke case, the two boundaries between regions 6 and 7 are tangent. Schefold's example is at a point, (φ t, σ t)=(1,1⁄2), in the thin wedge for region 6 in Figure 2. I did not find that points in the parts of region 6 in previous posts had more visually compelling wage frontiers than the point that Schefold found

Figure 2: Partitions of the Parameter Space Sixth 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.
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.
6	Gamma, Alpha, Gamma, Alpha	Around the second switch point, a lower rate of profits is associated with a LESS round-about technique and LOWER output per worker.
7	Gamma, Alpha	Around the switch point, a lower rate of profits is associated with a more round-about technique and greater output per worker.

The partitions of parameter space show that two values of σ t can be found as functions of φ t, where the corresponding wage curves are tangent at a switch point. Figure 3 plots the rate of profits and the wage for the switch points for these combinations of parameters. One set of three switch points is shown as a solid line and the other as a dashed line. The non-repeating switch point, for each set, is not a fluke except when on an axis or at the extreme right. The switch points for each set of parameters converges to a single switch point, with an increasing φ t. The convergence is complete at the double-fluke case.

Figure 3: Rate of Profits and the Wage at Certain Fluke Switch Points

The History Of No-Longer-Existing Socialism Validates Marx

Marx, like Adam Smith and Walt Rostow, had a stages theory of history. Feudalism was succeeded by capitalism, and capitalism is to be succeeded by socialism. Socialism is to arise first in the most advanced capitalist countries. (The theory of history is not my favorite part of Marxist theory.)

Russia, in 1917, was a semi-feudal country with peasants as the largest class. I guess China was the same, before Mao. A Marxist would not expect socialism to be successful in either country.

I think Lenin and the Bolsheviks agreed with this thesis when they first came to power. They expected their revolution to kick off revolutions elsewhere in Europe. And their expectations seemed to be initially met, what with the Spartacist uprising in Germany, Hungary, and so on.

Lenin, knowing that Russia was not ripe for socialism, talked about state capitalism even before the October revolution. Stalin invented the doctrine of socialism in one country. Economic development in the USSR and, I guess, in China, was amazing, albeit with much brutality. But eventually, further development required some semblance of capitalism

Is this not just what a Marxist would expect?

References

• Steve Paxton. 2021. Unlearning Marx [I have barely started this].
• Terry Eagleton. 2011. Why Marx was Right.

A Fourth And Fifth Double-Fluke Switch Point For A Triple-Switching Example

Figure 1: Partitions of the Parameter Space Around the Double-Fluke Switch Points

This post is a continuation of this series of posts.

The fourth and fifth double-fluke cases, in order of an increasing φ t, are symmetrical. The fourth case has two switch points between Alpha and Gamma. One is on the wage axis. The wage curves are tangent at the other switch point, at a positive rate of profits below the maximum. Alpha is cost-minimizing at all feasible rates of profits. Gamma is cost-minimizing only at the switch points. The fifth case also has two switch points between Alpha and Gamma. One is on the axis for the rate of profits, and the wage curves are tangent at the other switch point. Gamma is cost-minimizing at all feasible rates of profits. Alpha is cost-minimizing only at the switch points.

Figure 1 illustrates the partitions of the parameter space around these two double-fluke cases. Region 6, in which triple-switching occurs, extends throughout the figure. A perturbation across a partition corresponding to a switch point at which wages curves are tangent removes two successive switch points along the wage frontier. Thus, two of the switch points in the triple-switching case are lost in region 7, and only one switch point exists for this region.

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.
6	Gamma, Alpha, Gamma, Alpha	Around the second switch point, a lower rate of profits is associated with a LESS round-about technique and LOWER output per worker.
7	Gamma, Alpha	Around the switch point, a lower rate of profits is associated with a more round-about technique and greater output per worker.

A Third Double-Fluke Case For A Triple-Switching Example

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

This post is a continuation of this series of posts.

The next double-fluke case to be considered arises for parameters on intersection of the upper and lower boundaries of regions 3 and 5. Figure 1 illustrates this case, while Figure 2 depicts local perturbations of this double-fluke case. Perturbations that lead to either of the switch points at the extremes of the rate of profits no longer being at a feasible rate result in reswitching, as in regions 3 and 5. One switch point, as in region 2, results from perturbations in which both fluke switch points no longer being at a feasible rate of profits. But consider a perturbation n which both switch points occur at a positive rate of profits below the maximum. This example is of triple-switching in region 6.

Figure 2: Partitions of the Parameter Space around the Third 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.
6	Gamma, Alpha, Gamma, Alpha	Around the second switch point, a lower rate of profits is associated with a LESS round-about technique and LOWER output per worker.

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.