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.

Some Works Of Mainstream Economics?

Apparently, many mainstream economists assert that anything worthwhile in economics will be published in one of a few journals. The following is a selection of some articles from these well-respected journals, as I understand it:

• American Economic Review
• Donald J. Harris. 1973. Capital, distribution, and the aggregate production function. AER. 63(1): 100-113.
• David Laibman and Edward J. Nell. 1977. Reswitching, Wicksell effects, and the neoclassical production function. AER, 67(5): 878-888.
• Economica
• Murray Milgate. 1976. On the origin of the notion of ‘intertemporal equilibrium’. Economica. New series 46(181): 1-10.
• Journal of Economic Literature
• G. C. Harcourt. 1969. Some Cambridge controversies in the theory of capital. JEL. 7(2): 369-405
• A. S. Eichner and Jan Kregel. 1975. Post-Keynesian economic theory: a new paradigm in economics? JEL. 13: 1293-1314.
• Amartya Sen. 2003. Sraffa, Wittgenstein, and Gramsci. JEL 41(4): 1240-1255.
• Journal of Economic Perspectives
• Paul Davidson. 1991. Is probability theory relevant for uncertainty? A Post Keynesian perspective. JEP 5(1): 129-143.
• Julie A. Nelson. 1995. Feminism and economics. JEP. 9(2): 131-148.
• Avi J. Cohen and G. C. Harcourt. 2003. Whatever happened to the Cambridge capital theory controversies? JEP 17(1): 199-214.
• Journal of Economic Theory
• Alain Lipietz. 1982. The so-called ‘transformation problem’ revisited. JET. 26(1): 59-88.
• Quarterly Journal of Economics
• Michael Bruno, Edwin Burmeister, and Eytan Sheshinski. 1966. The nature and implications of the reswitching of techniques. QJE. 80(4): 526-533.
• Paul A. Samuelson. 1966. A summing up. QJE. 80(4): 568-583.
• John Eatwell. 1975. Mr. Sraffa’s standard commodity and the rate of exploitation. QJE. 89(4): 1975.
• Review of Economic Studies
• Joan Robinson. 1953-1954. The production function and the theory of capital. RES. 21(2): 81-106.
• Pierangelo Garegnani. 1970. Heterogeneous capital, the production function and the theory of distribution. RES. 37(3): 407-436.
• Review of Economics and Statistics
• Anwar Shaikh. 1974. Laws of production and laws of algebra: the humbug production function. REandS. 56(1): 115-120.

What I get out of this is that much of what is taught in mainstream microeconomics and macroeconomics is without theoretical and empirical foundation. Alternatives, such as Post Keynesianism, exist. Karl Marx's work is of interest to modern economists. These results were established decades ago.

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