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

The History Of No-Longer-Existing Socialism Validates Marx

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

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

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

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

Is this not just what a Marxist would expect?

References

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

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

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

This post is a continuation of this series of posts.

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

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

Table 1: Cost-Minimizing Technique byRegion

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

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

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

This post is a continuation of this series of posts.

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

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

Table 1: Cost-Minimizing Technique byRegion

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

Why Is Marginalist Economics Wrong?

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

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

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

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

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

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

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

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

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

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

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

Selected References

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

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

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

This post is a continuation of this series of posts.

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

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

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

Table 1: Cost-Minimizing Technique byRegion

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

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

Machinery And The Honesty Of David Ricardo

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

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

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

Edinburgh 5 June 1821

My dear Sir

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

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

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

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

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

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

ever faithfully yours

J. R. McCulloch

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

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

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

Edit: Reference as suggested in coments:

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

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.