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.

Review of Boettke, Candela, and Truitt: The Socialist Calculation Debate

Boettke, Candela, and Truitt (2024) is in the 'Cambridge elements: Austrian economics' series. It is 85 pages

The first chapter is an overview. Von Neumann, Kantorvich, and Leontief are mentioned. The authors acknowledge that current scholarship rejects the revisionism of Don Lavoie, which seemed to have won around the time of the fall of the Soviet Union. Chapter 2 is a recap of the original debate. Chapter 3 presents the views of Hayek, including in rebuttal of Lange and Lerner. Hayek's The Road to Serfdom is emphasized, including some negative reviews. Chapter 4 goes into Lavoie's revisionism and some socialist responses. Chapter 5 is mostly the authors' response to cybersocialist proposals. They say contemporary scholars miss the point. Tacit knowledge of time and space does not even exist without a price system. Chapter 6 is about economists you might not realize were inspired by the calculation argument. This includes Buchanan and public choice, Coase and transaction costs, and Alchian and evolutionary arguments. Kirzner is also mentioned, but you would expect him. Chapter 7 is a two-page conclusion.

I do not understand this book series. Entries in it are short and cannot be comprehensive. This book does not mention Barone's paper, before Von Mises. It does not mention any literature or arguments distinguishing Hayek's and Von Mises' arguments. Instead, Hayek is treated as if his argument is continuous with Von Mises. It is, I think, unclear on welfare comparisons. I know that Von Mises or Hayek were supposedly clear that they were not making claims about maximizing a social welfare function or about Pareto optimality. I do not know why David Ramsay Steele is not referenced.

I appreciate the references, but I do not think one can get a clear understanding of the arguments against Von Mises or Hayek. Some of the references were new to me. Trying to explain general equilibrium theory (GET) shortly is a challenge. The importance of the socialist calculation argument can be seen in motivating developments in GET. I also do not see how you can understand more current arguments about computational complexity from this book

I am not sure one can fully understand Von Mises or Hayek's arguments, either. The authors accurately notes that Von Mises grants the planners knowledge of the prices of consumer goods, as coming from a market just for them (pp. 17-18). And, on p. 24, Von Mises grants the planner 'complete information about the technological possibilities of their era'. As far as I am concerned, Von Mises' argument is invalid.

But, on p. 24-25, the authors take it back. They say, Mises had his argument 'under constant refinement in articulation of the details.' Supposedly, Hayek's emphasis on disequilibrium is also in Von Mises, specifically, in his argument about socialist calculation.

Strangely enough, this book has Hitler as a socialist, along with Stalin and Mao. The book echoes Hayek's assertion that socialists created the intellectual climate for Nazis. The historical dubiousness of this claim is not noted. Following Hirschman, I think The Road to Serfdom is a jeopardy argument. I agree with the authors that it is not an argument about a slippery slope, a kind of perversity argument.

I think this book must be directed to those who already have quite a background. It does not have the space to back up its assertions with a scholarly apparatus. Maybe it is for those who already have quite a background, but want to find out more about 21st-century arguments, particularly the reactions of some economists of the Austrian school to them. The book has no index.

References

• Barone, Enrico. 1908, 1935. The ministry of production in the collectivist state. (Tr. in Collectivist Economic Planning (ed. by F. A. Hayek).
• Bockman, Johanna. 2011. Markets in the Name of Socialism: The Left-Wing Origins of Neoliberalism. Stanford University Press.
• Boettke, Peter, Rosolino A. Candela, and Tegan L. Truitt. 2024. The Socialist Calculation Debate. Cambridge University Press.
• Steele, David Ramsay. 1999. From Marx to Mises: Post Capitalist Society and the Challenge of Economic Calculation. Open Court.