Even More On Recurrence Of Truncation Without Reswitching

Figure 1: Partitions in a Parameter Space with Small a1, 1, Small a1, 3

1.0 Introduction

I have been exploring patitions of parameter spaces by fluke switch points. I always have trouble visualizing higher dimensional spaces. Sometimes, my examples are two-dimensional. If I were doing more formal mathematics, instead of numerical exploration, I would not need to care.

This post is a continuation of the example developed here, here, and here. I present two-dimensional slices of a four-dimensional space.

2.0 Perturbations of Coefficients of Production

As the result of technical change, coefficients of production vary. No variation in labor coefficients or of the output matrix are considered here so as to retain reverse labor substitution at the switch point between Beta and Delta and forward substitution of labor at the switch points between Alpha and Gamma and between Gamma and Delta. Accordingly, consider perturbations of the coefficients of production in the first row of the input matrix. These parameters define a four-dimensional space. How does the analysis of the choice of technique vary with a decrease in these coefficients? The recurrence of truncation turns out be only a transient possibility in secular time with this specific model of technical change. The reverse substitution of labor can occur without the recurrence of truncation, but is also transient. Capital reversing, also transient, occurs in one region of parameter space.

2.1 Small Amounts of Circulating Capital Needed for New Machines

The parameter space is partitioned by parameters corresponding to fluke switch points. Figure 1 shows partitions with the values of a_1,1 and a_1,3 as in this previous post. 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. Four of the five partitions shown correspond to a switch point on the axis for the rate of profits. With four techniques, switch points between six pairs are possible. A switch point between Alpha and Delta can only occur as an intersection of all four curves. The same goes for a switch point between Beta and Gamma. These two pairs of techniques correspond to the partition for the switch point at which all four wage curves intersect, called here a four-technique pattern.

The sequence of techniques along the wage frontier is invariant in each numbered region. Table 1 lists the cost-minimizing techniques in each region, in order of an increasing rate of profits. Consider perturbations of the coefficients of production, a_1,2 and a_1,4, that specify the inputs of corn needed for each process in which the machine is run for the second year of its life. In region 1, to the northeast, old machines require so much circulating capital to operate that it is never cost-minimizing to run them for a second year. The Alpha technique is cost-minimizing, whatever the distribution of income. An improvement in the efficiency of old machines in using circulating capital in the machine industry leads to machines being operated for two years in that industry at large rates of profits, as in region 5. A similar improvement in the efficiency of old machines in the corn industry leads to machines being operated for two years in the corn industry, as in region 2. The recurrence of truncation only occurs in region 4.

Table 1: Overview of Regions

Region	Techniques	Notes
1	Alpha	No switch point.
2	Alpha, Gamma	Lower rate of profits associated with truncation in corn industry, greater output per worker.
3	Alpha, Gamma, Delta	Lower rate of profits associated with truncation, greater output per worker.
4	Alpha, Gamma, Delta, Beta	Recurrence of truncation in corn industry.
5	Alpha, Beta	Lower rate of profits associated with truncation in machine industry, greater output per worker.
6	Gamma, Delta	Lower rate of profits associated with truncation in machine industry, greater output per worker.
7	Gamma	No switch point.
8	Beta	No switch point.
9	Delta	No switch point.
10	Delta, Beta	Lower rate of profits associated with extension of economic life in corn industry, reverse substitution of labor.
11	Gamma, Delta, Beta	Lower rate of profits associated with truncation in machine industry, extension of economic life in corn industry, reverse substitution of labor.
12	Alpha, Beta, Delta	Around the Beta vs. Delta switch point, lower rate of profits associated with truncation, smaller output per worker.
13	Alpha, Beta, Delta	Lower rate of profits associated with truncation, greater output per worker.

2.2 Large Amounts of Circulating Capital Needed for New Machines in the Corn Industry

Figure 2 shows the partitions in the parameter space at a higher level of a_1,3. The structure in Figure 1 has moved upwards and to the left. The wedge in which the recurrence of truncation appears, region 4, has widened a bit. A new partition has appeared, for a fluke switch point between Gamma and Delta on the wage axis. For parameters where the partitions between regions 2 and 3 and between regions 2 and 7 intersect, the wage frontier has fluke switch points on both the wage axis and the axis for the rate of profits. For a low enough value of a_1,4, it is no longer cost-minimizing to truncate the machine in both industries at a low rate of profits. Gamma is cost-minimizing, whatever the distribution of income, for a high enough value of a_1,2 and a low enough value of a_1,4.

Figure 2: Parameter Space with Small a1, 1, Large a1, 3

2.3 Large Amounts of Circulating Capital Needed for New Machines in the Machine Industry

Figure 3 shows the partitions in the parameter space at a higher level of a_1,1. The amount of corn needed to operate a new machine in the corn industry, a_1,3, is the same as in Figure 1. Only the partitions and regions to the right are labeled in the figure. This part of the parameter space resembles those slices examined above, with a couple of variations. To the right, the lower boundary of region 5 corresponds to a switch point on the axis for the rate of profits, not a fluke switch point in which all four wage curves intersect. The upper boundary of region 3 now is, to the right, such a four-technique pattern of switch points.

Figure 3: Parameter Space with Large a1, 1, Small a1, 3

Figure 4 is an enlargement of the lower left of this slice of the parameter space. Three new points that are a fluke in multiple ways have appeared. One point corresponds to the intersection of the four wage curves on the wage axis. This point is the intersection of five partitions in the parameter space, just as the point in parameter space corresponding to the intersection of the four wage curves on the axis for the rate of profits. The region in which the recurrence of truncation occurs, region 4, has appeared. The reverse substitution of labor also occurs in regions 10 and 11. Two points in the part of the parameter space depicted correspond to fluke switch points simultaneously lying on both axes.

Figure 4: Parameter Space with Large a1, 1, Small a1, 3 (Enlarged)

Figure 5 enlarges this slice of the parameter space in the middle of Figure 3. The sequence of cost-minimizing techniques, in order of an increasing rate of profits, is Alpha, Beta, and Delta in both region 12 and region 13. The boundary between regions 12 and 13 corresponds to an intersection of the Beta and Delta wage curves on the waxis but not on the frontier. In region 12, the wage curves for Beta and Delta intersect twice in the first quadrant. The second intersection is on the frontier. It is a switch point exhibiting capital-reversing. Around the switch point between Beta and Delta a lower rate of profits is associated with a decreased capital-intensity and decreased net output per worker in the economy as a whole.

Figure 5: Parameter Space with Large a1, 1, Small a1, 3 (Another Enlargement)

3.0 Conclusion

The exploration of partitions in parameter space supports an analysis of technical change in which the circulating capital needed to operate a machine decreases. A trajectory from the upper right to the lower left in the figures and from Figures 2 or 3 to Figure 1 represent this specific form of technical change.

This analysis illustrates how perturbing coefficients of production affects the analysis of the choice of technique. Parameter spaces are partitioned by fluke switch points into regions in which the variation of the cost-minimizing technique with distribution is itself invariant. Some of the structures formed by intersections of partitions are not specific to the example, or even to models of fixed capital. For example, fluke switch points can arise in many models simultaneously on both the wage axis and the axis for the rate of profits. Perturbations of parameters around such a point eliminates or creates the possibility for certain techniques to be cost-minimizing at extreme ranges for the rate of profits.

Likewise, perturbation analysis supports the analysis of specific forms of technical progress. By contrast with fluke switch points, the recurrence of truncation, the reverse substitution of labor, and capital-reversing are not flukes. The reverse substitution of labor occurs in regions in which the recurrence of truncation does not occur. In this sense, the reverse substitution of labor is more general in this example. Capital-reversing arises in a region in which neither of these other 'perverse' phenomena occur. Configurations of parameters in which these phenomena arise, however, define only a few regions of the parameter space. This observation seems capable of generalization to other 'perverse' phenomena in other examples. Other forms of technical progress can be explored.

On Confusion About Gödel's Theorem, Including In Austrian Economics

1.0 Introduction

Gödel's theorems are often invoked in non-mathematical contexts, sometimes in a very imprecise fashion (Franzen 2005, Raatikaine 2007, Jaimungal). You should be skeptical of what many say about Gödel, including of what I say.

In this post, I look at two examples. One is of a confused economist of the Austrian school. The other is of Wittgenstein, who has been ably defended on this point.

2.0 A Claim About the Economic Calculation Problem

Nguyen (2024) says that somehow Gödel's theorem supports the claim that centralized economic planning is, in principle, impossible. Nguyen is sympathetic to the claim that our minds (brains?) transcend the capabilities of all formal systems. I find this claim dubious, but I have not read Penrose. For purposes of argument, Nguyen assumes, through most of this paper, that the central planner has the information von Mises grants them. Nguyen deliberately puts aside Hayek's concerns about non-articulated, distributed, tacit knowledge.

I have demonstrated that von Mises' argument is invalid. I find I am not original. Cockshott (2010) has done the same. Both of us put forth a linear programming formulation that does not require prices of intermediate goods as part of the data. We differ in our specifications of the planner's objective function. Neither of us are echoing Lange and Lerner's formulations of general equilibrium. As such, I do not see any issues are raised for us by the non-computability of utility functions, preference relations, or general equilibria in which ratios of marginal utilities enter the system of equations.

A polynomial-time algorithm exists for solving linear programs. Linear programs are not undecidable. So Gödel's theorem and issues arising from Turing's work do not seem to have any purchase here.

Furthermore, in practice, only a finite number of numbers would be used in drawing up plans. Real numbers would be approximated, if you can call it that, by IEEE Std. 754. This standard defines floating-point numbers, in single and double precision formats. Your computer does not only fail to represent the full range of real numbers. It also only represents a finite number of integers in words, typically of 32 or 64 bits.

I do not mean to suggest that many practical issues do not arise with central planning Nor am I advocating such. I continue to maintain that von Mises' argument is invalid. And I find Nguyen's paper to be one of those mystical, imprecise invocations of Gödel.

3.0 A Notorious Paragraph from Wittgenstein

Here is Wittgenstein from notes not published until after his death:

"8. I imagine someone asking my advice; he says: 'I have constructed a proposition (I will use "P" to designate it) in Russell's symbolism, and by means of certain definitions and transformations it can be so interpreted that it says: "P is not provable in Russell's system". Must I not say tha this proposition on the one hand is true, and on the other hand unprovable? For suppose it were false; then it is true that it is provable. that surely cannot be! And if it is proved, then it is proved that it is not provable. Thus it can only be true, but unprovable.'

Just as we ask, '"Provable" in what system?,' so we must also ask, '"True" in what system?' 'True in Russell's system' means, as was said, proved in Russell's system, and 'false in Russell's system' means the opposite has been proved in Russell's system. - Now what does your 'suppose it is false' mean? In the Russell sense it means, 'suppose the opposite is proved in Russell's system'; if that is your assumption you will now presumably give up the interpretation that it is unprovable. And by 'this interpretation' I understand the translation into this English sentence. — If you assume that the proposition is provable in Russell's system, that means it is true in the Russell sense, and the interpretation 'P is not provable' again has to be given up. If you assume that the proposition is true in the Russell sense, the same thing follows. Further: if the proposition is supposed to be false in some other than the Russell sense, then it does not contradict this for it to be proved in Russell's system. (What is called 'losing' in chess may constitute winning in another game.)" -- Wittgenstein (1978: Part I, Appendix III)

Wittgenstein seems to equate provability in PM with being true in PM. Is not part of Gödel's point to separate these concepts? Furthermore, Wittgenstein seems to be writing only about the heuristic argument given at the start of Gödel's paper. Do his remarks make sense of Gödel numbering, (primitive) recursive functions, and so on?

As I understand it, Gödel's proof of his incompleteness theorem is a conventional proof. The proof is an argument to convince you that a sequence of statements exists that follow one another, by conventional deduction rules. And the conclusion is a theorem about natural numbers.

Gödel's proof is about the syntactic manipulation of strings of symbols by formal rules. Wittgenstein questions interpretations, I guess, of Godel numbers as the statements or sequence of statements that map into them. Floyd and Putnam (2000) justify Wittgenstein in a way that draws on the distinction between consistency and ω-consistency.

J. B. Rosser replaced ω-consistency with consistency in Gödel's proof. A theory is ω-inconsistent if one can show, for some proposition p:

• not p(0), not p(1), not p(2), ..., and
• There exists x such that p(x).

I guess that a non-standard model can be ω-inconsistent, where x is not a natural number, somehow. Suppose PM is ω-inconsistent. And suppose the negation of the Gödel sentence 'P' is true. Then the Godel number for the proof of this proposition could be a non-standard natural number. Wittgenstein writes about interpretations, but does not mention ω-consistency. I suppose he could have known about the Löwenheim-Skolem theorem.

I am sympathetic to being suspicious of English-language interpretations of syntactical manipulations. I am not so sympathetic to how lightly Wittgenstein treats possible contradictions in mathematics. I am sympathetic to the idea that mathematical logic, set theory, and model theory, for example, just provides more maths. One does not need this math to justify what humans have been doing for millennia. Do mathematical propositions have meaning before proofs are found? As I understand Wittgenstein, he says not. Proofs draw connections and give the proved proposition a meaning.

But I am also aware that even if I can echo out some propositions from these fields, I am no expert in them.

References

• Cockshott, W. Paul. 2010. Von Mises, Kantorovich and in-natura calculation. European Journal of Economics and Economic Policies 7(1): 167-199.
• Dawson, Ryan. 2015. Leaving Mathematics As It Is: Wittgenstein's Later Philosophy of Mathematics. University of East Anglica. PhD Thesis.
• Floyd, Juliet and Hilary Putnam. 2000. A note on Wittgenstein’s 'notorious paragraph' about the Gödel theorem. Journal of Philosophy. 97(11): 624-632.
• Franzen, Torkel. 2005. Gödel's Theorem: An Incomplete Guide to Its Use and Abuse. A. K. Peters.
• Jaimungal, Curt. Theory of Everything: a podcast episode.
• Nguyen, Hai-Trieu. 2024. The incompleteness of central planning. Quarterly Journal of Austrian Economics. 27(4).
• Raatikaine, Panu. 2007. Review of Franzen. Notices of the AMS. 54(3): 380-383.
• Standford Encylopedia of Philosophy, Wittgenstein's philosophy of mathematics.
• Wittgenstein, Ludwig. 1978. Remarks on the Foundation of Mathematics, revised edition. MIT Press.

Technical Change and Triple-Switching in the Corn-Tractor Model

I have another working paper, Technical change and triple-switching in the corn-tractor model, at the Centro Sraffa. The abstract follows.

Abstract: With triple-switching, each of two techniques are cost-minimizing in two disjoint intervals of the wage or rate of profits. Technology that supports multiple switch points between two techniques can only be a temporary phenomenon, as one technique supplants another with technical progress. A perturbation analysis of a triple-switching example in the corn-tractor model illustrates this claim. A parameter space, defined by two selected coefficients of production, is partitioned by loci corresponding to fluke switch points. The analysis of the choice of technique does not qualitatively vary within each of the resulting regions. Technical progress corresponds to specific trajectories through this parameter space.

Keywords: Cambridge Capital Controversy; Fixed Capital; Reswitching of techniques

JEL Codes: B51: Sraffian; C67: Input-Output Models; D24: Production, Cost, Capital

Howard Fast Leaves The Communist Party Of The USA After Khrushchev's Secret Speech

This post is another instance of me stumbling over anti-communist literature, in a sense.

I read Howard Fast's novel, Citizen Tom Paine, many years ago. I have never read Spartacus or seen the movie. I think I read April Morning in high school. I think I read Freedom Road at some point. At any rate, I have the impression that reconstruction was actually a moment when it seemed America was coming closer to living up to its promise to provide liberty for all. I also read The Last Frontier at some point.

I believe I knew he was blacklisted. I hadn't known that he actually was a member of the CPUSA. Nor had I known about Peekskill or his time in prison.

Nikita Khrushchev gave his secret speech in February 1956. In it, he denounces the cult of personality around Stalin. I believe this is where the phrase comes from. It looked like that the Soviet Union might fulfill its promise. But by the end of that year, the Soviet Union invaded Hungary to put down their revolution.

The secret speech caused Fast all sorts of struggle. When he left the communist party, he wrote a book explaining his reasons.

"I had a god who walked naked, but nobody among those I loved said so; for even as the innocent wisdom of Hans Christian Andersen held that those who could not see the king's clothes were persons of small intellect and unfit for the positions they held, so in my world, it was the conviction of millions of good and wise folk that only those who had lost all honor, dignity, decency and courage would dare to point out that this god whom we worshiped for his noble raiment was indeed naked and ugly in his nakedness.

Who would be without honor, dignity, decency and courage?

In the little town where I live, there is a little store, unimportant and of no consequence, and out of this store an old man ekes a living. This is an old man who mourns a hurt which will not heal, the kind of hurt many who read this will know intimately, for twenty years ago the young son of this man fell in Spain, fighting in the Lincoln Battalion for the Spanish Republic and the freedom of men. His son lies buried in the distant Spanish soil, and for twenty years the hurt in this old man was as if it had happened yesterday.

He had a little salve to rub on the terrible sore. This was the salve, that his son had died in the best of causes, the fight for the liberation of mankind. But in 1956, a man called Khrushchev delivered a certain 'secret report' - telling a story of Russia and the Communist movement that I and my friends had heard before but had never believed before. Now Khrushchev made proof of twenty'five years of 'slander,' and we believed. And among those who believed because they had to believe was this old man whose son had laid down his life in Spain.

I came into his store one day in that month of June and he was weeping. He asked me,

'Why did my son die?'

For had I not held, all of my thinking life and in all that I wrote, that one son of man was all the sons of man? He then said to me, but not in words - for a broken heart does not make a gentle person cruel or vindictive - not in words but with the look in his eyes -

'That I, a plain man did not comprehend this is no wonder; but you, Howard Fast, spoke and wrote and pleaded this cause - and why? Can you tell me why?'" -- Howard Fast, The Naked God: The Writer and the Communist Party, 1957.

None of this means Fast did not keep the same values:

"I am neither disillusioned nor depressed, and I have lived through grim times, but times when mankind made gigantic strides forward. Though the Communist Party is disciplined and often splendid in military action, I do not think it can claim credit for the events we have seen. Socialism, justice and the brotherhood of man are mighty and irresistible forces; they will grow to fruition in spite of the Communist Party - and Soviet socialism will not forever lie supine under the heel of the commissar."-- Howard Fast, chapter 15.

Capital-Reversing In A Pertubation Of An Example Of The Recurrence Of Truncation

Figure 1: The Wage Frontier for an Example of Capital-Reversing

I am continuing to explore perturbations of coefficients of production for inputs of circulating capital in this example. The example is of the recurrence of truncation without either the reswitching of techniques or capital reversing. This post presents a perturbation in which the recurrence of truncation no longer occurs, but capital-reversing now arises.

Tables 1 and 2 present the coefficients of production for the example. The technology has the minimum structure, in a model in which multiple commodities are produced, and inputs of labor, circulating capital, and fixed capital are needed in each industry. Also, managers of firms in both industries have a choice of the economic life of a machine. In this perturbation, a_{1, 1} is higher than in the example I started with. a_{1, 3} is the same. I varied a_{1, 2} and a_{1, 4} to find this example of capital-reversing.

Table 1: Inputs for The Technology

Input	Industry
	Machine		Corn
	I	II	III	IV
Labor	1/10	8	43/40	1
Corn	32/50	2/5	1/8	0.38944767
New Machines	1	0	1	0
One-Year Old Machines (1st type)	0	1	0	0
One-Year Old Machines (2nd type)	0	0	0	1

Table 2: Outputs for The Technology

Output	Industry
	Machine		Corn
	I	II	III	IV
Corn	0	0	1	14/25
New Machines	2	5/2	0	0
One-Year Old Machines (1st type)	1	0	0	0
One-Year Old Machines (2nd type)	0	0	1	0

Table 3 repeats the definition of the available techniques. I will observe that a switch point between Beta and Gamma cannot exist without that switch point being simultaneously a point at which all wage curves intersect.

Table 3: Specification of Techniques

Technique	Processes
Alpha	I, III
Beta	I, II, III
Gamma	I, III, IV
Delta	I, II, III, IV

Figure 1, at the top of this post, depicts the wage curves for the four techniques. It is not visually striking. In particular, the wage curves for Beta and Delta are hard to distinguish by eye. Figure 2, below, is an enlargement of part of the wage curves so that you can see that Beta and Delta have different wage curves. The sequence of wage curves on the frontier, in order of an increasing rate of profits, is Alpha, Beta, Delta. Gamma is never on the frontier.

Figure 2: The Wage Frontier for an Example of Capital-Reversing (Enlargement)

This example is one of my usual illustrations that the Austrian and marginalist theories of capital are muddled and confused. Around the switch point between Beta and Delta, a lower interest rate is associated with the truncation of the machine in the corn industry and a less-capital intensive technique. Net output per worker has decreased for the economy as a whole. Around the switch point between Alpha and Beta, a lower interest rate is associated with the truncation of the machine in the machine industry and a more capital-intensive technique. Net output per worker is increased for the economy as a whole.

A lower interest rate need not encourage capitalists to adopt more capital-intensive techniques. No necessary association exists with extending the economic life of a machine and adopting a more capital-intensive technique.

Socialism Worked In A Village In China In 1979

By socialism, I mean 'socialism with Chinese characteristics'. An emphasis on developing and liberating the forces of production is one aspect of socialism. Trying to seek truth from facts is one way that you might phrase one of those Chinese characteristics. Another characteristic is a matter of seeking democratic initiatives from below, especially from rural areas. The principle of household responsibility is in tension with the principle of collectively 'eating from one big pot'. But Mao's 'On contradiction' shows that such tensions will continue in socialism. Household responsibility is not in tension with a community collectively owning the land.

This inadequate preamble suggests why socialists could embrace these events:

"On the 24th of November, 1978, representatives from the 18 families of Xiaogang Village, of Fengyang County in Anhui Province, met and signed what was then a secret document. In 79 characters, the document stated that each family would subdivide their collective land, work their allocated plots to meet government quotas, and then sell any surplus for their own benefit. The reason: back in 1958 the village population was 120, but 67 died from hunger during 1958–1960 (in the midst of the 'Great Leap Forward'). Starvation had haunted them once again in 1978 and they feared for the future. The result of the secret agreement: in the following year, the farmers of Xiaogang village produced six times the amount of grain compared to the previous year, and the per capita income of the farmers increased from 22 to 400 RMB. Why was the document a secret? With the fully collectivised system in force, any form of buying and selling was regarded as a 'capitalist' exercise and thus punishable. The farmers knew they were taking a risk, but they were fortunate that the local and provincial CPC officials were sympathetic to their endeavour. So also was the new leadership of the country, with Deng Xiaoping at the head. By the next spring, the word of Xiaogang's move was out. While some accused them of undermining socialism, the country's leadership saw it very differently: this would be the beginning of the household responsibility system and thus of the rural reform that drove the first period of the Reform and Opening-Up. By 1984, the household responsibility system had been implemented across the country." – Roland Boer. 2021. Socialism with Chinese Charateristics: A Guide for Foreigners. Springer. p. 85

I certainly do not think of socialism as a blueprint to be fashioned beforehand and imposed from above. Any feasible development of socialism will include the development and modification of institutions and policies at different times and places. The Reform and Opening-Up initiative seems to have been a good idea at the time, although maybe, like the French Revolution, it is too soon to tell. Later developments showed the need for a tack more towards port.

Boer's book is in definite contrast with David Harvey's 2005 A Brief History of Neoliberalism. Harvey is one of those foreigners who Boer says do not understand China. I think of the events recounted above as not too far from how I understand the German mark community or Russian mir village assembly worked. After these events, the land remained collectively owned. Neither absentee owners nor wage laborers existed. The existence of markets is not sufficient for capitalism, as can be seen by millennia of pre-capitalist experience with simple commodity production.

Boer depicts the cultural revolution, starting in 1966, and the gang of four, as deviations from Mao Zedong Thought. He depicts the inequality growing in the 'wild 90s' as a deviation, as well. Boer ties some ideas in the Confucian tradition to some Marxist ideas. The time of xiaokang (moderate or acceptable time of well-being) precedes datong (great unity). Likewise socialism precedes Communism. Boer writes about the four moderizations, in agriculture, industry, national defence, and science and technology, which China has been bringing about in the last few decades. According to Boer on Mao, contradictions will still exist in socialism, but they will not necessarily be antangonistic. Boer has a schematic approach to contradiction analysis. You should identify the principal contradiction and pay attention to the particularity of the contradiction.

When Boer writes about two systems in one one country, he writes about the Hong Kong Special Administration Region (SAR), the Macau SAR, and Taiwan, as if Taiwan was not a different country. He says nothing about 1989 events in Tienanmen Square. In writing about human rights, he emphasizes that China is an example of anti-colonialism, while claims that human rights are individual and innate is a claim that came from the western peninsula of Europe. I had not thought of China in this context. I am more likely to think of Stephen Biko and Frantz Fanon like this. I know very little about China.

Boer, in his discussion of Xi Jiping thought, emphasizes this 2018 speech on Marx. It is confusing to think how a country can be run by communists who introduce institutions that are widely perceived as capitalist, and yet the ruling party still perceives themselves as building communism.

A Two-Commodity Example Of Harrod-Neutral Technical Change And The Choice Of Technique

Figure 1: Wage Curves for a Fluke Case

1.0 Introduction

This post revisits Harrod-neutral technical change, in the context of the choice of technique. I used Matlab code to obtain the results in this post.

2.0 Technology

The technology (Table 1) in this example is a modification and extension of a reswitching example from Bruno, Burmeister, & Sheshinski (1966). Each column specifies the physical inputs (labor-hours, tons, bushels) needed to produce a specified output at a specified point in time. Each process uses up its inputs and exhibits constant returns to scale (CRS). Technical change results in the reduction of labor coefficients. The labor coefficients for the first processes in the two industries decrease at the same rate, while the labor coefficients in the other two processes also decrease at the same rate, but possibly differing from the rate of decrease in the first processes.

Table 1: A Technology

	Iron Industry		Corn Industry
	Process a	Process b	Process c	Process d
Labor	e1 - θt	(2/5) e1 - φt	(33/50) e1 - θt	(1/100) e1 - φt
Iron	a1,1,a = 0	a1,1,b = 1/3	a1,2,c = 1/50	a1,2,d = 71/100
Corn	a2,1,a = 1/10	a2,1,b = 1/20	a2,2,c = 3/10	a2,2,d = 0
OUTPUTS	1 ton iron	1 ton iron	1 bushel corn	1 bushel corn

A technique consists of a process for producing corn and a process for producing iron. Four techniques (Table 2) exist in this economy. Iron and corn are basic commodities, in the sense of Sraffa, in all techniques. Alpha and Delta experience Harrod-neutral technical change, possibly at different rates. Beta and Gamma combine processes from the two techniques with neutral technical change. How does the analysis of the choice of technique, with prices of production, vary as technical change occurs in secular time?

Table 3: Specification of Techniques

Technique	Iron Industry	Corn Inudstry
Alpha	Process a	Process c
Beta	Process a	Process d
Gamma	Process b	Process c
Delta	Process b	Process d

3.0 Prices of Production and the Choice of Technique

Prices of production must be such that managers of capitalist firms are willing to continue producing iron and corn. In calculating prices of production, I abstract from secular change in labor coefficients.

A price system is associated with each technique. I assume labor is advanced, and wages are paid out of the surplus at the end of the year. A bushel corn is the numeraire. For each technique, the wage and the price of iron can each be expressed as a function of the rate of profits.

Figure 1 plots the wage curves for the four techniques, at a given time and with given rates of technical change. The technique with the highest wage is cost-minimizing at a given rate of profits. The outer envelope is the wage frontier. It is composed out of the wage curves for the Delta, Alpha, and Beta techniques. The wage curve for Gamma is on the frontier only at the first switch point, at approximately 45.8 percent. This is a fluke switch point. Around this switch point, a higher wage or lower rate of profits results in processes in both industries changing in the cost-minimizing case. Only one industry has two processes in cost-minimizing techniques in a non-fluke switch point.

The other switch point, at approximately 167.1 percent, is not a fluke and illustrates capital- reversing. The Beta technique requires less labor, through the economy as a whole, to produce a net output of a bushel corn than the Alpha technique does. Around the switch point, a higher wage is associated with firms wanting to employ more labor per bushel corn produced net in the economy.

4.0 Partitions of the Parameter Space by Fluke Switch Points

A switch point at which four wage curves intersect is only one of four fluke switch points that arise in the example, depending on the rates of technical progress and the time. Figure 2 shows a partitioning of the parameter space, based on these fluke switch points. Each of the partitions is an affine function with a slope of unity. This property of the partitioning of the parameter space, that all partitions are straight parallel lines with unit slope, is a consequence of considering techniques with processes drawn from two techniques undergoing Harrod-neutral technical change.

Figure 2: A Partition of the Parameter Space

The analysis of the choice of technique is qualitatively invariant in each numbered region. Table 3 lists the cost-minimizing techniques, in order of an increasing rate of profits, in each region. Only techniques on the frontier are listed. Capital-theoretic ‘paradoxes’ that arise in each region are noted. Only the switch point in region 5 corresponds to obsolete marginalist intuition. A lower wage or higher rate of profits, around the switch point, is associated with a more labor- intensive and less capital-intensive technique.

Table 3: Specification of Techniques

Region	Techniques	Properties
1	Delta, Gamma, Alpha, Beta	Capital-reversing, reverse substitution of labor for Alpha vs. Beta switch point.
2	Delta, Beta, Alpha, Beta	Reswitching, capital-reversing, and the reverse substitution of labor.
3	Beta, Alpha, Beta	Reswitching, capital-reversing, and the reverse substitution of labor.
4	Alpha, Beta	The reverse substitution of labor.
5	Alpha, Beta	'Non-perverse' switch point.

5.0 A Trajectory through the Parameter Space

The dashed line in Figure 2 represents a possible trajectory through the parameter space, with fixed rates of technical progress for the production processes. Figures 3 and 4 graph the maximum wage and the wage at switch points, as functions of time, along this trajectory. At the intersection of the trajectory with the partition for the fluke switch at which at which four wage curves intersect the wage curve for the Beta technique replaces the wage curve for the Gamma technique on the frontier. The wage curve for the Delta technique no longer appears on the frontier at a nonnegative rate of profits after the trajectory passes the partition for the switch point between Beta and Delta on the wage axis. Similarly, when the trajectory crosses the next partition, the first switch point between Alpha and Beta is no longer on the frontier at a nonnegative rate of profits.

Figure 3: A Trajectory Through the Parameter Space

Figure 4: A Trajectory Through the Parameter Space (Cont'd)

When this switch point between Alpha and Beta exists at a rate of profits of -100 percent, a_c,0,2 = a_c,0,2. In other words, around the other switch point, at a positive, feasible rate of profits, a higher wage leaves unchanged how much labor is hired per unit of gross output in the corn industry. Before the trajectory crosses the partition for this fluke switch point, the labor coefficient in the last process is less than the labor coefficient in the penultimate process. A higher wage around the illustrated switch point is associated with more employment in the corn industry per unit of gross output. This is the reverse substitution of labor.

Harrod-neutral technical progress cannot change the ranking of techniques by the maximum rate of profits. In this example, one of the mixed techniques, Beta, is cost-minimizing at the maximum rate of profits. At the start, Delta is cost-minimizing at a rate of profits of zero, and the example exhibits capital-theoretic ‘paradoxes’. If the rate of neutral technical progress in Alpha exceeds the rate of neutral technical progress in Delta, Alpha must eventually be cost-minimizing at a rate of profits of zero. The trajectory in the example illustrates how capital-theoretical ‘paradoxes’ can disappear.

6. Conclusions

Harrod-neutral technical progress yields particularly simple structures in the parameter space. All partitions corresponding to fluke switch points are parallel affine functions with slopes of unity, and the rates of profits at which fluke switch points occur do not vary with neutral technical progress. The fluke at which four wage curves intersect illustrates this property. No double fluke cases, at the intersections of partitions can arise here. Likewise, no fluke switch points can appear on the axis for the rate of profits.

I previously claimed that technology that supports multiple switch points between two techniques can only be a temporary phenomenon, as one technique supplants another with technical progress. The results of the numerical experimentation in this post are in tension with that claim. In some trajectories, neutral technical change eliminates the capital-theoretic ‘paradoxes’ of reswitching and capital-reversing. In other trajectories, it does not.

I created the example to start with reswitching. If neutral technical progress in Delta exceeds that in Alpha, the reswitching example persists through secular time.

In practice, technical change will vary in its rate and have biases. For example, Marx-biased technical change is a mix of capital-using and labor-saving technical change (Foley, Michl, & Tavani 2019). Technical change will often involve more than processes from two existing techniques. It frequently includes the creation of new industries and new commodities. The analysis of Harrod-neutral technical change, entangled with the choice of technique, provides a baseline to contrast with structures in parameter spaces found in other analyses.