Perturbations Of Selected Parameters In The Corn-Tractor Model

Figure 1: Partitioning a Part of the Parameter Space with Fluke Cases

1.0 Introduction

This post is a continuation of the first example here. I examined a perturbation of two parameters of that example. I ended up with a more perspicacious partition of the parameter space than here.

2.0 Technology

Table 1 merely repeats the parameters for the fluke case that I started with. This case has switch points on the axis for the rate of profits and on the wage axis. A third switch point exists at an intermediate rate of profits.

Table 1: Parameters for Technology for First Example

Parameter	Type I Tractors	Type II Tractors
Tractor input per tractor (a)	≈ 0.306226	2/5
Labor input per tractor (b)	≈ 233.6967	20
Years tractors last in tractor industry (n)	1	2
Tractor input per bushel corn (α)	1	20
Labor input per bushel corn (β)	αI bI/aI	850
Years tractors last in corn industry (ν)	1	2

3.0 Perturbations of Selected Parameters

Almost any perturbation of the model parameters destroys fluke properties of the example in the previous section. Figure 1 illustrates perturbations in a_II and α_II. A switch point is on the axis for the rate of profits only for a specific value of a_II. Likewise, a switch point is on the wage axis only for the depicted partition of the parameter space, of, for instance, regions 1 and 7. The example in the previous post has parameters found at the intersection of these two partitions. The two other partitions occur for parameter values at which a switch point is repeated and the two wage curves are tangent at this switch point. The regions bounded by these partitions of the selected part of the parameter space are numbered.

The dashed line depicts the combination of coefficients of production for which the ratio of labor to tractors does not vary between industries for tractors of type II. To the left and above this line the physical capital-intensity of production is less, for type II tractors, in producing new tractors than it is in producing corn. To the right and below, the tractor industry for type II tractors has a larger physical capital-intensity than corn production.

I would have liked to have drawn the partitions as three-dimensional manifolds in a four-dimensional space, where (a_II, b_II, α_II, β_II) is a point in the space. But I can visualize a tesseract only momentarily, if at all (Heinlein 1941). Figure 1 is constructed by selecting only two parameters to perturb.

Double-fluke cases arise at intersections of the partitions. The partition between regions 2 and 3 is tangent to the partition between regions 3 and 4 at their point of intersection. Similarly, the partition between regions 1 and 5 is tangent to the partition between regions 2 and 5. The two partitions between regions 6 and 7 are tangent at their point of intersection, as well.

This last double-fluke switch point can perhaps admit of some elaboration. Figure 2 shows the rate of profits and the wage at switch points for each of two fluke cases. The solid lines correspond to the partition between regions 1 and 5 and the lower partition in Figure 2 between regions 6 and 7. The dashed lines correspond to the partition between regions 3 and 4 and the upper partition. Three switch points exist for the parameters along these two partitions. Two of these switch points are repeated roots, which is the fluke case under consideration. All three switch points coincide on the wage frontier at the double-fluke switch point to the extreme left.

Figure 2: The Rate of Profits at Switch Points with Tangent Wage Curves

Table 2: Cost-Minimizing Techniques by Region

Region	Cost-Minimizing Technique	Notes
1	Type II	No switch point. Type II tractors are dominant with sufficiently low coefficients of production in producing Type II tractors.
2	Type II, Type I	Around the switch point, a lower rate of profits is associated with a more roundabout technique, a greater capital-output ratio, and more consumption per person-year.
3	Type I, Type II, Type I	Around the first switch point, a lower rate of profits is associated with a less roundabout technique, a higher capital-output ratio, and more consumption per person-year. Around the second switch point, a lower rate of profits is associated with a more roundabout technique, a lower capital-output ratio, and less consumption per person-year.
4	Type I	No switch point. Type I tractors are dominant with sufficiently high coefficients of production in producing Type II tractors.
5	Type II, Type I, Type II	Around the first switch point, a lower rate of profits is associated with a more roundabout technique, a greater capital-output ratio, and more consumption per person-year. Around the second switch point, a lower rate of profits is associated with a less roundabout technique, a lower capital-output ratio, and less consumption per person-year.
6	Type I, Type II, Type I, Type II	Around the first and third switch point, a lower rate of profits is associated with a less roundabout technique, a greater capital-output ratio, and more consumption per person-year. Around the second switch point, a lower rate of profits is associated with a more roundabout technique, a smaller capital-output ratio, and less consumption per person-year.
7	Type I, Type II	Around the switch point, a lower rate of profits is associated with a less roundabout technique, a greater capital-output ratio, and less consumption per person-year.

The analysis of the choice of technique is qualitatively invariant in each numbered region. Table 2 lists the cost-minimizing technique, in order of an increasing rate of profits, in each region. One technique is cost-minimizing, whatever the distribution of income, in regions 1 and 4. One switch point exists in regions 2 and 7. Double-switching occurs in regions 3 and 5. Finally, triple-switching occurs in region 6. Perturbations of the parameters for an example in the previous post can result in each type of tractor being cost-minimizing in two discrete ranges of the wage or the rate of profits. This partitioning is not unique to this model.

4.0 Conclusions

I can examine specific properties of the switch points in each region, and maybe draw some more conclusions. But that will be for future posts.

Duncan Foley On Why General Equilibrium Maybe Is Not Neoclassical Economics

Duncan Foley participated in a 2003 conference comparing and contrasting general equilibrium theory with long period models developed by advocates of the Cambridge capital critique. This is from the wrap-up discussion on the last day.

"I am in a somewhat peculiar position because due to certain idiosyncrasies of my education I never learned 'neoclassical economics'. The economic theory that I learned wasfrom Herbert Scarf and it was couched entirely in terms of the abstract general equilibrium model with n commodities, abstract production sets, and so forth. Someone has defined economic intuition as what you learn in your first course in economics, and since I did not learn the same things as many other people, my intuition is not the same. My economic intuition was always that there were no theorems of the following kind: suppose you increase the supply of labour, as a result the equilibrium real wage will fall. I knew there were no such theorems available in general equilibrium theory. I also knew, and maybe this became clearer because of Scarf's mathematical point of view, that once you took the step to a simultaneous general equilibrium vision, you had to give up hope of sustaining some traditional ideas. For one thing, you lost any sense of causality. In a general equilibrium model there is no real sense in which one factor causes another: everything is determined simultaneously by the whole collection of relationships... And I never thought there was any hope of proving general stability theorems, because Herbert Scarf had showed that you could not prove stability, at least tatonnement stability, without unacceptably strong hypotheses on aggregate demand functions. So these things are not counterintuitive to me, these are just facts of the matter. I see them as inherent defects of the general equilibrium point of view. I tend to read this as a case where neoclassical economics, in an attempt to purify itself logically, leached out all of the substance, the concrete substance and content, of what it had to say about the world. And it raises the question in my mind whether it is really fair to talk about general equilibrium theory as if it were neoclassical economics, precisely because it does not have that content of substitutability, well-behaved stability properties and well-behaved comparative statics properties. In fact, when I was at MIT it became apparent to me that for, say, Solow and Samuelson, who are perhaps closest to being true believers in the old neoclassical rules, general equilibrium theory was just as much a threat as the Cambridge controversy to their point of view." -- Duncan Foley (2003) final discussion, in: General Equilibirum: Problems and Propsoects (ed. by Fabio Petro and Frank Hahn), London: Routledge.

I have not looked at the American Economic Review, for example, in quite some time. But I have long lost an ability to make sense of mainstream economics. Doubtless, market power of employers, search costs for employees in finding new jobs, montoring costs and principal agent problems for employers, and the suggestion that an employee is a 'lemon' if they offer to work for a lower wage than many are all of empirical importance. But why must they be invoked to explain persistent unemployment or the failure of a rise in the minimum wage to create disemployment? The rigorous theory, in both the comparison of intertemporal general equilibrium paths and in comparisons of long period positions does not yield downward-sloping demand functions for labor. Likewise, rigorous theory does not yield regular supply and demand relations for other markets either.

Do mainstream economists even have an articulated, overall vision for microeconomics? Neither general equilibrium theory or game theory seem to provide one. Is it sufficient to just have lots of little models? This seems to be the position of mainstream economics for decades.

Double Fluke Cases For Triple-Switching In The Corn-Tractor Model

Figure 1: Wage Curves for an Example With Tractors Lasting One and Two Years

1.0 Introduction

This post presents two examples in the corn-tractor model. These examples are double fluke cases. Each has three switch points. One is on the wage axis, and another is on the axis for the rate of profits. Perturbations of parameters of each example can result in triple-switching.

The corn-tractor model is a fixed capital model, an adaption of the Samuelson-Gargenani model. The consumption good, corn, can be produced by labor working with any one of a number of different types of tractors. Each type of tractor is produced by labor with an input of that type of tractor. Each type of tractor lasts for a specified number of years in the production of new tractors and of corn. Its lifetime can vary between industries, and these lifetimes can vary among types of tractors. This is an example of joint production. Every process for producing a new tractor, except the last, also produces tractors one year older than the tractors used as inputs. The production of corn also yields a joint product of tractors one year older. Each type of tractor works with constant efficiency, whether in producing new tractors or in producing corn. With these assumptions, no choice of the economic life of a machine arises. The tractor will be used for its full physical life in each industry.

2.0 An Example with One and Two-Year Old Tractors

A technique is identified with a type of tractor. Six parameters (Table 1) specify a technique. The numerical example consists of a choice between two types of tractors. The first lasts only one year. That is, the production and operation of the first type of tractor is an example of circulating capital. The second type lasts two years in both the production of new tractors and of corn. The ratio of labor to tractors does not vary between industries for the first type of tractors. In other words, physical capital-intensity does not vary between industries. The production of corn is more capital-intensive than the production of new tractors for the initial parameters for the second type of tractors. As Steedman (2019) notes, this special case is sufficient to yield triple-switching.

Table 1: Parameters for Technology for First Example

Parameter	Type I Tractors	Type II Tractors
Tractor input per tractor (a)	≈ 0.306226	2/5
Labor input per tractor (b)	≈ 233.6967	20
Years tractors last in tractor industry (n)	1	2
Tractor input per bushel corn (α)	1	20
Labor input per bushel corn (β)	αI bI/aI	850
Years tractors last in corn industry (ν)	1	2

I chose the parameters in Table 1 to illustrate a double-fluke case. The parameters for type II tractors are arbitrary, but such that the convexity of the corresponding wage curve changes once along its length. The convexity cannot vary more than once for tractors that last two years. The parameters for type I tractors are constrained to provide switch points on the wage axis and the axis for the rate of profits. These constraints result in a knife-edge case in which certain perturbations of parameters result in triple-switching.

Figure 1, at the top of the post, illustrates the wage curves in this case. They are hard to see by eye. Type II tractors are cost-minimizing at high wages, low positive rates of profits. Type I tractors are cost-minimizing at low positive wages, high rates of profits.

For what it is worth, I also include a graph (Figure 2) of the variation in the capital-output ratio, with the rate of profits, in this example. In a stationary state, tractors of each age are operated in parallel, both in the tractor industry and in the corn industry. After each year, the oldest tractors are discarded and the appropriate number of new tractors are added to the stock in each industry. The sum of the prices of production of these tractors is the value of capital. Following Steedman, I take a non-physical measure of capital-intensity to be the ratio of the value of capital to the value of net output. The capital-output ratio is a dimensionless number, while the units for the ratio of the value of capital to employment depends on the choice of the numeraire.

Figure 2: Capital-Output Ratio an Example With Tractors Lasting One and Two Years

In the numerical example, the capital-output ratio is a constant, independent of the rate of profits, for type I tractors. It increases and then decreases, with the rate of profits, for type II tractors. Since a switch point exists on the wage axis, the capital-output ratio does not vary, with the type of tractors, at the two switch points with a positive rate of profits. In the jargon, real Wicksell effects are zero at these switch points (Harris 1973). Around the switch point at approximately 45 percent, a lower rate of profits, is associated with the adoption of a more roundabout technique, even though this increases, across stationary states, neither the capital-output ratio nor consumption per worker. I here identify roundaboutness with the number of years a tractor lasts.

3.0 An Example with One and Three-Year Old Tractors

I also created a double-fluke case (Table 2) for an example in which one type of tractors lasts one year, and the other type lasts three years. Figure 3 shows the wage curves for this case. Figure 4 is the corresponding graph for the capital-output ratio.

Table 2: Parameters for Technology for Second Example

Parameter	Type I Tractors	Type III Tractors
Tractor input per tractor (a)	≈ 0.2391377	31/100
Labor input per tractor (b)	≈ 82.723374	7
Years tractors last in tractor industry (n)	1	3
Tractor input per bushel corn (α)	1	21
Labor input per bushel corn (β)	αI bI/aI	400
Years tractors last in corn industry (ν)	1	3

Figure 3: Wage Curves for an Example With Tractors Lasting One and Three Years

Figure 4: Capital-Output Ratio for an Example With Tractors Lasting One and Three Years

4.0 Conclusion

This post has validated Steedman's claim that triple-switching can arise in the corn-tractor model as he claims. In both examples, one type of tractor lasts for one-year. In that circulating capital case, the process for producing more tractors is as capital-intensive as the process for producing corn. The corresponding wage curve is a straight line, the price of new tractors does not vary with the rate of profits, and the capital-output ratio is also constant.

In both examples, the other type of tractor lasts more than one year. It lasts the same amount of time in producing new tractors and in producing corn. Tractors operate with constant efficiency over their lives in both industries. Consequently, the price of an old tractor of a specified age is the same in each industry. Production of the consumption good is more physcially capital-intensive than production of capital good. By this, I mean the ratio of tractors (of a given age) to labor is greater in the corn industry than in the tractor industry. Consequently, the price of new tractors of the second type varies with the rate of profits, and non-zero price Wicksell effects exist.

Nevertheless, these examples do not have visually appealing wage frontiers. Perturbing parameters will show that my prior claims about how parameter spaces are partitioned are qualitatively replicated here.

Reference

• Gargenani, Pierangelo. 1970. Heterogeneous capital, the production function and the theory of distribution. Review of Economic Studies 37 (3): 407-436.
• Samuelson, Paul A. 1962. Parable and realism in capital theory: the surrogate production function. Review of Economic Studies 29 (3): 193-206.
• Steedman, Ian. 2020. Fixed capital in the corn-tractor model. Metroeconomica 71: 49-56.

Gunnar Myrdal Sounding Like Tony Lawson?

This passage suggests to me that, in economics, one cannot expect to find event regularities from surface level data:

"The really important difference between us and our natural science colleagues is illustrated by the fact that we never reach down to constants like the speed of light and of sound in a particular medium, or the specific weights of atoms and molecules. We have nothing corresponding to the universally valid measurements of energy, voltage, amperes, and so on. The regularities we find do not have the firm, general, and lasting validity of 'laws of nature.'

If we economists, for instance, establish by observation the income or price elasticity for, say, sugar, our findings are valid for only a specific group of consumers in a single community or region at a particular time - not to mention the fact that the concept elasticity itself loses what I call adequacy to reality, and thereby analytical usefulness, in underdeveloped countries that have no, or very imperfect, 'markets,' in the sense given to this term by the economists." -- Gunnar Myrdal, Against the Stream: 138-139.

Myrdal wrote a lot about methodology. He was intererested in how unacknowledged valuations enter into economic theory. And he thinks social scientists should explicitly state their valuations. But he does not write about ontology.

I don't know how this applies to me. I suppose you can say that my focus on distribution, especially wages, reflects some valuations. I think, though, that I am mostly focusing on mathematics. And by looking for structures in parameter spaces for open models of prices of production, I am not making claims about event regularities at surface levels. I leave to others to relate movements in market prices to prices of production. Really, though, when I first learned about Robinson and Sraffa, I was astonished that a serious reason exists to think intermediate microeconomics, as widely taught, is nonsense, not even wrong.

The Emergence of Triple Switching and the Rarity of Reswitching Explained

I have written up a series of post as a research paper: first post, second, third, fourth, fifth, sixth, seventh. Here I present the abstract and most of the introduction.

Abstract: Empirical research indicates that the reswitching of techniques, as well as multiple switching with more switch points, is rare. This article explores parameter spaces in the analysis of the choice of technique to suggest why reswitching and triple-switching might be hard to find in empirical data. An example illustrates that the emergence of triple-switching requires specific evolutions of coefficients of production. Further evolution of technology removes the possibility of triple-switching. The example also illustrates that the roundaboutness of a technique is independent of the capital-intensity of a technique.

Introduction

Consider the analysis of the choice of technique in post-Sraffian price theory. Kurz & Salvadori (1995) is a standard textbook presentation. Switch points, in which two techniques are both cost-minimizing at a given wage or rate of profits, are found as the zeros of certain polynomials of high degree. These zeros can be complex and, if real, need not be positive and below the maximum rate of profits. Nevertheless, theory suggests that multiple switch points between techniques are common. Han & Schefold (2006) and Zambelli (2018) are the most comprehensive empirical works to date, looking at switch points in comparing techniques drawn from Leontief matrices constructed from actual national income and product accounts. Reswitching and capital reversing, never mind multiple switching with more switch points, seem to be rare in empirical data. How can this discrepancy between expectations from theory and empirical results be resolved?

Kurz (2020) points out some difficulties with the empirical results. Often fixed capital is not taken into account. Only circulating capital is assumed, and the production of heterogeneous commodities, with varying input coefficients, in each industry is abstracted from. Some of these heterogeneous processes in an industry can be expected to be obsolete in the year in which data is gathered. Obsolete plant is operated in an economy side-by-side with more recent vintages. Firms often produce multiple products, and accounting conventions may assign a firm to different industries in different years. Heterogeneity in labor, changes in labor mixes, and changes in relative wages over time, are also ignored in this empirical work. The empirical research to date, although impressive still suffers from limitations that ought to be taken into account when assessing how rare reswitching is likely to be.

Nevertheless, Schefold (2023) investigates the supposed rarity of certain capital-theoretic phenomena, found surprising by marginalist economists. He randomly generates coefficients of production for alternate techniques. The resulting wage curves are near linear, that is, nearly affine functions. A small number of techniques, only one or two, contribute their wage curves to the frontier, except near extremes for the rate of profits. The continuous variation in the cost-minimizing technique with distribution, as postulated in marginalist theory, does not seem defensible. The reswitching of techniques does not seem likely on the wage frontier.

Changes of techniques in practice seem not to be a matter of choosing a cost-minimizing technique from an existing and well-known book of blueprints, following price signals. Rather, as Joan Robinson frequently remarked, new techniques are a matter of technical innovation, with reduced coefficients of production and perhaps with processes using new capital goods, not previously produced.

This article explores parameter spaces for technology with a different method. An example of triple-switching from Schefold (1980), to illustrate roundabout production, is extended with technological change. This particular model of structural economic dynamics (Pasinetti 1993) is not claimed to be realistic. Rather, it provides a two-dimensional parameter space that is partitioned by fluke switch points. A switch point is a fluke if it is a knife edge case in which almost all perturbations of model parameters destroy its defining properties. This article identifies points in the parameter space that are double-fluke cases. For instance, the wage curves at such a point are tangent at a switch point that is also on the wage axis. Each double-fluke case occurs for parameters that are intersections of two partitions in the parameter space. A picture of how triple-reswitching can arise emerges from a synthesis of local perturbations around these double-fluke cases. This extension of the analysis of the choice of technique suggests why triple-switching, for example, might be hard to find in empirical data. The example illustrates that the emergence of triple-switching requires specific evolutions of coefficients of production. Further evolution of technology removes the possibility of triple switching.

Two Sad Stories About Great Mathematicians

Here is a story about David Hilbert torwards the end of his career:

"Otto Neugebauer, now an associate professor, was placed at the head of the Mathematical Institute. He held the famous chair for exactly one day, refusing in a stormy session in the Rector's office to sign the required loyalty declaration. The position of the head of the Mathematical Institute passed to Weyl. Although his wife was part Jewish, he was one of those who thought that something might yet be salvaged. All during the bitter uncertain spring and summer of 1933 he worked, wrote letters, interviewed officials of the government. But nothing could be changed.

By late summer nearly everyone was gone. Weyl, vacationing with his family in Switzerland, still considered returning to Göttingen in the hope that somehow he could keep alive the great scientific tradition. In America, his many friends worried about him and wrote long letters, advising, urging, begging that he leave Germany before it was too late. Abraham Flexner offered him a position at the Institute for Advanced Study. Finally Einstein, who had already been at the newly created Institute for several years, prevailed upon the younger man to come and join him there.

In Göttingen, Hilbert was left almost alone. He kept Bernays on as his assistant at his own expense. The Foundations of Mathematics, which he and Bernays had written in collaboration, was almost ready for publication. He put away his general mathematical books and became progressively more distant. With Bernays's help, he saw Arnold Schmidt and Kurt Schütte through the doctorate. Schütte was the last of 69 mathematicians (40 of them during the years from 1900 to 1914) to receive their degrees from Hilbert. In actuality, however, all of Schütte's contacts were through Bernays. He saw Hilbert only once.

'When I was young,' Hilbert said to young Franz Rellich, one of the few remaining members of the old circle, 'I resolved never to repeat what I heard the old people say - how beautiful the old days were, how ugly the present. I would never say that when I was old. But, now, I must.'

Sitting next to the Nazis' newly appointed minister of education at a banquet, he was asked, 'And how is mathematics in Gottingen now that it has been freed of the Jewish influence?'

'Mathematics in Göttingen?' Hilbert replied. 'There is really none any more.'" -- Constance Reid. 1996. Hilbert

Here is Kurt Gödel becoming an American citizen:

"Morgeristern had many stories to tell about Gödel. One concerned the occasion when, in April 1948, Gödel became a U.S. citizen, with Einstein and Morgenstern as witnesses. Gödel was to take the routine citizenship examination, and he prepared for it very seriously, studying the United States Constitution assiduously. On the day before he was to appear, Gödel came to Morgenstern in a very excited state, saying: 'I have discovered a logical-legal possibility by which the U.S.A. could be transformed into a dictatorship.' Morgenstern realized that, whatever the logical merits of Gödel's argument, the possibility was extremely hypothetical in character, and he urged Godel to keep quiet about his discovery at the examination. The next morning, Morgenstern drove Gödel and Einstein from Princeton to Trenton, where the citizenship proceedings were to take place. Along the way Einstein kept telling one amusing anecdote after another in order to distract Gödel, apparently with great success. At the office in Trenton, the official was properly impressed by Einstein and Morgenstern, and invited them to attend the examination, normally held in private. He began by addressing Gödel: 'Up to now you have held German citizenship.' Gödel corrected him, explaining that he was Austrian. 'Anyhow', continued the official, 'it was under an evil dictatorship... but fortunately, that's not possible in America.' 'On the contrary,' Gödel cried out, 'I know how that can happen!!' All three had great trouble restraining Gödel from elaborating his discovery, so that the proceedings could be brought to their expected conclusion." -- Solomon Feferman. 1986. Gödel's life and work. In Kurt Gödel Collected Works: Volume I. Oxford University Press.

I wish these stories had no current relevance. I suppose it is encouraging of what others in the past had to overcome.

Recap For A Triple -Switching Example

Figure 1: Actual and Stylized Partitions of Parameter Space with Triple-Switching

This post is a continuation of this series of posts.

The partitioning of the parameter space by fluke switch points in these posts can be combined into one picture. The left pane in Figure 1 illustrates. The dashed line is a ray from the origin, discussed below. I find this complete picture for this example hard to perceive by eye. The right pane provides a highly stylized presentation of the partitions, rotated and stretched. The partitions are not straight lines on the left. The boundary between regions 1 and 5 is tangent to the boundary between regions 1 and 2 at the point of intersection. The boundary between regions 3 and 4 is likewise tangent to the boundary between regions 2 and 4 where they intersect. The two boundaries between regions 6 and 7 become tangent at their point of intersection. Region 6 is for triple-switching. It adjoins regions 3, 5, and 7. Regions 3 and 5 are examples of reswitching. Their borders with region 6 have fluke switch points on the axis for the rate of profits and on the wage axis, respectively. The borders between regions 6 and 7 are associated with fluke switch points in which two wage curves are tangent. Reswitching in regions 3 and 5 appears in a fairly generic fashion. Reswitching can also appear from perturbations of coefficients of production, where the region in parameter space corresponding to reswitching is not adjacent to a region in which triple-switching occurs.

Table 1: Cost-Minimizing Technique by Region

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 example demonstrates that an increase in the roundaboutness of the cost-minimizing technique is independent of its capital-intensity. For the traditional marginalist story, a lower rate of profits around a switch point is associated with a choice of technique with greater capital-intensity and greater output per worker. For the traditional story from the Austrian school, a lower rate of profits around a switch point is associated with the adoption of a more roundabout technique.

Yet all four entries in the grid in Table 2 are populated by switch points in the example. Consider the switch point in region 2 or the first switch point in region 5. Around these switch points, a lower rate of profits is associated with the adoption of a more capital-intensive, but a LESS roundabout technique. The less roundabout and more capital-intensive technique has a greater output per worker. These switch points populate the lower left entry in the table. The second switch point in region 5 populates the upper right entry. Around this switch point, a lower rate of profits is associated with the adoption of a more roundabout technique with LOWER output per worker. The first switch point in region 3 and the switch point in region 7 populate the upper left entry. They happen to be consistent with these old theories. The second switch point in region 3 fills the entry in the lower right in the table. Roundaboutness and capital-intensity move together, but against the intuition of outdated marginalist and Austrian school economists. A lower rate of profits is associated with a LESS round-about technique and LOWER output per worker. The first and third switch points in region 6, in which triple-switching occurs, are like the switch point in region 7. The second switch point is like the second switch point in region 3. Roundaboutness and capital-intensity do not seem to have much to do with one another.

Table 2: Lower Rate of Profits around a Switch Point

	Traditional Marginalist Story	'Perverse' Marginalist Story
Traditional Austrian Story	Greater net output per worker	Smaller net output per worker
	More roundabout technique	More roundabout technique
'Perverse' Austrian Story	Greater net output per worker	Smaller net output per worker
	Less roundabout technique	Less roundabout technique

Figure 2: Structural Dynamics for an Example of Triple-Switching

These posts explore structures in parameter spaces that might not be immediately visible in empirical regularities at surface levels. Since distribution is not specified, the model of the choice of technique is open. The impact on the dynamics of market prices of coefficients of production supporting triple-switching is not clear. Such temporal dynamics, one might expect, depend on the speed with which capitalists adopt processes adapted to new technology and distribution, as compared to the speed with which technology improves. Market dynamics might depend on the history of such adjustments, as reflected in fixed capital remaining from previous adjustments. The size of the extra profits obtainable by these adjustments is another consideration. Even if triple-switching were quickly manifested in struggles over the distribution of income and in market dynamics, the partitioning of parameter spaces by fluke switch points suggests that triple-switching might be rare. It only occurs in specific examples of structural dynamics.

These posts demonstrate that triple-switching can arise through innovations in technology. The illustrated traversal of the parameter space is not the only way. Reswitching can arise as here and as adjacent to a triple-switching example. Likewise, triple-switching can arise adjacent to an instance of quadruple-switching. One can see this by considering generalizations of Figure 2. Each instance with more switch points is less likely to correspond to a region in the parameter space formed by coefficients of production or related parameters. At any rate, the number of partitions in parameter space increases, and their configurations are more complicated. In the example, triple-switching arises from technological innovation. But further innovation in the same direction removes the possibility of triple-switching. This result applies to reswitching, and generalizes to quadruple-switching, and so on. Regions with multiple switch points are transient, arising as one technique replaces another as dominant, whatever the distribution of income.

The example examined in the main text has also demonstrated that the degree of roundaboutness is independent of the capital-intensity of a technique. Keynes had a point:

"It is true that some lengthy or roundabout processes are physically efficient. But so are some short processes... Moreover there are all sorts of reasons why various kinds of services and facilities are scarce and therefore expensive relatively to the quantity of labour involved. For example, smelly processes command a higher reward, because people will not undertake them otherwise. So do risky processes. But we do not devise a productivity theory of smelly or risky processes as such." -- Keynes (1936)

This series of posts re-iterates that the rate of profits is not an index for the relative scarcity of capital. A lower rate of profits need not be associated with a technique that is either more capital-intensive or more roundabout. Likewise, the wage is not an index for the relative scarcity of labor.

Previous research suggests that perturbations in relative markups can also bring about the same variations in the analysis of the choice of technique as those that result from perturbations in coefficients of production (Vienneau 2024a). Hence, triple-switching also seems to be possible as a result of long-lasting variations in relative markups.