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.

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.