Prices In An Example Of The Life Of A Machine

Figure 1: Structural Economic Dynamics of the Price of a New Machine

I am not sure that there is any great insight here. But this post depicts the evolution of prices in an example which I have been explaining in two posts. For some parameters, I have found that a shorter economic life of a machine may be associated with a more capital-intensive technique around a switch point. This can occur at both a 'normal' and a 'perverse' switch point. This finding raises difficulties for tradional expositions of Austrian capital theory.

In this post, I repeat the specification of technology. Table 1 presents coefficients of production. Corn is both a consumption good and a capital good, insofar as it is an input into all four processes. Corn is also the numeraire. Machines are only capital goods. Technology is not yet fully specified in this example, with numerical values undefined for two coefficients of production. The machine does not work with a constant efficiency throughout its physical life. The inputs of seed corn needed to work with the machine increase in agriculture as the machine ages. Labor inputs first increase and then decrease. A one-year old machine is more efficient than both a two-year old and a three-year old machine. Whether a two-year old machine is more efficient than a three-year old machine cannot be determined based on physical data alone. I assume constant returns to scale and free disposal of old machines.

Table 1: Technology

Input	Industry	Agriculture
	I	II	III	IV
Labor	a0,1	39/200	117/200	39/100
Corn	a1,1	39/100	141/250	117/200
New Machines	0	1	0	0
1-Year Old Machines	0	0	1	0
2-Year Old Machines	0	0	0	1
Output
Corn	0	1	1	1
New Machines	1	0	0	0
1-Year Old Machines	0	1	0	0
2-Year Old Machines	0	0	1	0

I defined the techniques in the previous posts. In the Alpha technique, the machine is run for one year. In Beta, it is run for two years. In Gamma, it is run for its full physical life of three years.

The evolution of the coefficients for production for labor and corn inputs in manufacturing the machine is as noted in Figure 1. I showed previously that, with these specific assumptions, time is partitioned by fluke switch points. Aside from extremes at a time t of 0 and 9, I chose three instants of time, each at the midpoint of one of these partitions. Figures 1 and 2 graph the price of a new machine, one-year old machine, and two-year old machine as functions of the rate of profits at these instants of time.

Figure 2: Structural Economic Dynamics of the Prices of Old Machines

As time goes on, the price of production of each vintage machine decreases at any given rate of profits with technical progress in the manufacture of machines. The maximum possible rate of profits, when 'workers live on air', also increases with technical progress.

At the start of time, the Gamma technique is cost-minimizing for any feasible rate of time. This is depicted by the top curve in Figure 1 and both panes of Figure 2. At the end of time, technical progress has made the Alpha technique cost-minimizing for any feasible rate of profits. See the bottom curve in Figure 1. Since the prices of one-year old and two-year old machines are zero for Alpha, no curves for t = 9 appear in Figure 2.

The choice of the economic life of the machine varies with the rate of profits at intermediate times. About when t = 3, the price of a two-year old machine turns negative at a high enough rate of profits at Gamma prices. The machine can still be used in this range if it disposed of after running for two years. At the start of the second year, it is one year old and has a positive price. The Beta technique is cost-minimizing at a high enough rate of profits. At the switch point in which the price of the two-year old machine is zero, the prices of new and one-year old machines are the same for Beta and Gamma techniques. This common price of machines is an illustration of the general result that prices are unique at switch points.

A time around t = 5.214 is an example of reswitching. The price of a one-year old machine is negative, at Gamma and Beta prices, at a rate of profits of zero. Thus, for a low enough rate of profits, the Alpha technique is cost-minimizing. The price of a one-year old machine turns positive at Gamma prices, with an increase in the rate of profits, before it turns positive for Beta prices. A first switch point exists between Alpha and Gamma at a low rate of profits. At a higher rate of profits, the price of a two-year machine becomes zero at Gamma prices. This is the switch point between Beta and Gamma noted in the previous paragraph above. Finally, the price of a one-year old machine becomes negative at Beta prices at a high rate of profits, and Alpha is once again cost-minimizing.

A time around t = 5.96 is a simpler case of reswitching, between Alpha and Gamma. Two switch points exist, and the price of a one-year old machine is zero at Gamma prices at those switch points. The price of a one-year old machine at Beta prices is always negative for all feasible rate of profits.

This analysis demonstrates how examining when prices of production become negative relates to the determination of the cost-minimizing economic life of a machine. At switch points, the price of some old machine is zero.

Depreciation charges fall out of this analysis. Depreciation is the difference in price, at a given rate of profits, of a machine between two years. In the case of a machine of constant efficiency, charges for the cost of machines vary as the value of an annuity lasting for the length of the life of the machine. Modeling the price of a machine as varying as radioactive decay is not a good choice. The example illustrates the more general case of non-constant efficiency. Strangely enough, the machine appreciates from the second year to the third year when the cost-minimizing choice of technique is to run the machine for its full physical life.

Elsewhere

Does Learning How To Think In Coding Transfer More Generally?

• Another YouTube Video in the above series. I suspect some of us are not fans of some of those interviewed, although you must admit they changed the world.
• John Michael Colón, Wobbly Economics - Part I. A tribute to Fred Lee and an explanation of why you are right to distrust economists.
• Shattering 'Market Theory', an old post on Daily Kos, reporting a discussion on a private email list of one of my numeric examples.
• Draft of Ederer, Goldsmith-Pinkham, and Jensen (2023) on the vicious fools on EJMR. slides. These links are probably temporary, and I am in the data.

A Letter From Marx To Engels In 1868 On The First Volume Of Capital

Over years, I have considered how Marx continues and differs from classical political economy. I have also documented some foreshadowings and outlines of the transformation problem. This is another letter in a series

In this letter, Marx alludes to prices of production and the transformation problem. Apparently, he thinks at this time that volumes 2 and 3 will be a single volume.

Here he sets out three points which he thinks are original to the first volume of Capital. The first is that he abstracts from rent, interest, and profits to the general form of surplus value. Previously, he told Engels this is one of the two best points in his book. His second point in which he thinks he is original, he also thinks is one of his best points: namely, the treatment of concrete and abstract labor activities. His third original point is how he treats both time and piece wages. This is another example of abstraction. I think Marx was, despite his differences, always aware of his debt to the Ricardian socialists. Maybe, he treats them elsewhere with more respect than he ever treats Proudhon after ensuring that they never again would be drinking buddies.

8 January 1868

Dear Fred,

Ad vocem Duühring. It is a great deal from this man that he gives almost positive acceptance to the section on 'primitive accumulation'. He is still young. As a follower of Carey he is in direct opposition to the FREETRADERS. Furthermore, he is a university lecturer, and therefore not displeased that Professor Roscher, who blocks the way for them all, is receiving some kicks. One thing in his description struck me very strongly. That is, as long as the determination of value by labour time is itself left 'undetermined', as it is with Ricardo, it does not make people SHAKY. But as soon as it is brought exactly into connection with the working day and its variations, a very unpleasant new light dawns upon them. I believe one reason that Duühring reviewed the book at all is malice against Roscher. Indeed it is easy to scent his anxiety that he might also be 'Roscher'ed. Curiously, the fellow has not detected the three fundamentally new elements of the book:

1. that in contrast to all previous political economy, which from the outset treated the particular fragments of surplus value with their fixed forms of rent, profit and interest as already given, I begin by dealing with the general form of surplus value, in which all these elements are still undifferentiated, in solution as it were;

2. that the economists, without exception, have missed the simple fact that, if the commodity has the double character of use value and exchange value, then the labour represented in the commodity must also have a double character; thus the bare analysis of labour sans phrase, as in Smith, Ricardo, etc., is bound to come up against the inexplicable everywhere. This is, in fact, the whole secret of the critical conception;

3. that for the first time wages are shown as the irrational outward form of a hidden relationship, and this is demonstrated exactly in both forms of wages: time wages and piece wages. (It was a help to me that similar formulae are often found in higher mathematics.)

As for Mr Dühring's modest objection to the determination of value, he will be astonished when he sees in Volume II how little the determination of value counts for 'directly' in bourgeois society. Actually, no form of society can prevent the labour time at the disposal of society from regulating production in ONE WAY OR ANOTHER. But so long as this regulation is not effected through the direct and conscious control of society over its labour time - which is only possible under common ownership - but through the movement of commodity prices, then things will remain as you so aptly described them already in the Deutsch-Französische Jahrbücher.

Ad vocem Vienna. I am sending you various Vienna papers (of which you must return to me the Neues Wiener Tagblatt which belongs to Borkheim, and keep the rest), from which you will see two things: firstly how important Vienna is at this moment as a market place, since there is new life there; and secondly the way the matter should be handled there. I cannot find the address of Prof. Richter. Perhaps you have Liebknecht's letter which gives it. If not, ask him to send it to you, and then dispatch the article direct to Richter, but not via Liebknecht.

It seems to me that Wilhelmchen is by no means ALTOGETHER bona fide. He (for whom I have had to find so much time to make good his asininity in the Allgemeine Augsburger, etc.,) has so far found no time even to mention publicly the title of my book or my name. He overlooks the affair in the Zukunft so as not to be put in the embarrassing position of sacrificing his own independent greatness. And there was also no time available to say a solitary word in the workers' paper (Deutsche Arbeiterhalle, Mannheim), which appears under the direct control of his friend Bebel. In short, it is certainly no fault of Wilhelmchen that my book has not been totally ignored. First, he has not read it (although to little Jenny he made fun of Richter, who thinks that he needs to understand a book before he can publicise it), and secondly, after he had read it or claimed to have read it, he has had no time, although he has time, since I got him Borkheim's SUBVENTION, to write letters twice weekly to Borkheim; although, instead of sending the shares to Strohn for the money, which was transmitted to him through me and obtained by my good offices, he asks for Strohn's address, in order to play his tricks with him directly, behind my back, and swamp him with epistles as he does Borkheim. In short, Wilhelmchen wants to make himself important, and in particular the public should not be distracted from its interest in Wilhelmchen. We must now act half as if we did not notice this, but still treat him with caution. As for his call to Austria, you cannot believe him until it has happened. And secondly if it should come to this, we shall not dissuade him, but IF NECESSARY, simply explain to him what I explained to him when he joined Brass's Norddeutsche, that, if he should compromise himself again, he will be, if necessary, publicly disavowed. I told him this, in the presence of witnesses, when he moved off to Berlin at that time.

I think you can send articles direct to the enclosed Neue Freie Presse (Vienna). The present joint owner, Dr Max Friedländer (Lassalle's cousin and deadly enemy), was the person for whom I acted as a correspondent for a longish period for the old Vienna Presse and for the Oder-Zeitung.

Finally, with regard to the Internationale Revue, Fox (who was sent to Vienna by an English paper to pay a visit and establish connections) asked me, from Vienna a few days ago, for a letter of introduction to Arnold Hilberg. I sent it to him, and at the same time told the said Hilberg in this letter that circumstances had prevented us writing, that we would do something this year, etc.

Fortnightly Review. Professor Beesly, one of the triumviri who secretly direct this paper, has told his special friend Lafargue (whom he constantly invites to dine at his house) that he is morally certain (it completely depends upon him!) that a review would be accepted. Lafargue would hand it in to him himself.

Ad vocem Pyat. In today's Times you will see the ADDRESS of the FRENCH DEMOCRATS about FENIANISM. (which appeared 4 weeks ago) and was sent in by Pyat. What has happened is this. The French government has launched an investigation (particularly visites domiciliaires at the homes of our correspondents in Paris) against the International Association as a société illicite. Ditto probably sent to the British government letters about FENIANISM written by our Dupont. Mr Pyat, who always ran down our 'Association' as non-revolutionary, Bonapartist, etc., is afraid of this TURN of things, and is swiftly seeking to give the appearance that he has something to do with the matter and is 'MOVING'.

Ad vocem Benedek: can I have the journal for A FEW DAYS? YOU have now proven yourself twice a prophet, firstly a tactical prophet (in the Sevastopol affair), and secondly a strategic prophet (in the Prussian-Austrian affair). But the sense of sensible men cannot predict the stupidities of which man is capable.

Ad vocem carbuncles. Consulted doctors. Nothing new. Everything which the gentlemen have to say indicates that one has to have private means to live in accordance with their prescriptions, instead of being a poor devil like me, poverty-stricken as a church-mouse. When you see Gumpert, you can tell him that I feel (up to THIS MOMENT that I write) a stinging prickle in my body, that is my blood. It seems to me that for this year I shall not be quite over the affair.

My COMPLIMENTS TO Mrs Burns.

Salut

Your

Moor

Technological Progress In Industry And The Life Of A Machine In Agriculture

Figure 1: Variation of Switch Points with Technological Progress in Industry

This post is an expansion on this post.

Technological progress in industry, in which the machine is produced, can be illustrated in Figure 1 in the previous post by a movement roughly from off the graph to the upper right to below the lower left. More concretely, suppose each of the two non-zero coefficients of production in the machine industry decrease at a constant rate of σ₀ and σ₁ respectively. The two coefficients of production thus vary with time as an exponential function.

Variation in the efficiency of the machine with age in agriculture is unaffected by this technological progress in industry. One can calculate prices of production, including switch points, at each instant of time. The machine does not work with a constant efficiency throughout its physical life. The inputs of seed corn needed to work with the machine increase in agriculture as the machine ages. Labor inputs first increase and then decrease. Whether a two-year old machine is more efficient than a three-year old machine cannot be determined based on physical data alone.

Figure 2: Variation of Switch Points and Wages with Technological Progress in Industry

Figure 1 plots the maximum rate of profits and the rate of profits for switch points against time, for specified parameters for technological progress. Figure 2 shows the corresponding plot with the wage as the ordinate. Progress in industry manifests in the variation in the economic life of the machine, as it is used in agriculture. At the start, the machine is used for its full three years, whatever the distribution of income. It is used for only one year, whatever the distribution, at the end of the technological progress depicted here. Switch points, including so-called ‘perverse’ switch points arise during this process, but eventually vanish.

The vertical partitions in the figures occur at times in which at least one switch point is a fluke. The illustrated fluke cases are for a switch point arising at a rate of profits of zero, switch points at the maximum rate of profits, a switch point in which three wage curves intersect, and a switch point in which two wave curves are tangent. The rate of decrease of corn inputs per produced machine has been carefully chosen to ensure the occurrence of the fluke case illustrated in Figure 2 in the previous post. Generally, the fluke case in which the switch point between Alpha and Beta occurs at the maximum rate of profits will occur before or after the fluke case in which a switch point between Alpha and Gamma occurs at the maximum wage. One cannot expect technology to evolve such that two fluke cases occur simultaneously.

A fluke case is such that almost any perturbation of model parameters disturbs its qualitative properties. Between fluke cases, the number and sequence of switch points along the wage frontier, as depicted in Figures 1 and 2, are invariant. This perturbation analysis illustrates that switch points with positive real Wicksell effects are not fluke cases. Likewise, associations of a shorter economic life of a machine with greater capital intensity or with a lower rate of profits are not fluke cases.

How Would Socialism Work?

On Another Topic, with an Appearance by Rutger Bregman.

This post does not answer the question, but merely provides a bibliography. I have not read everything below. I suppose this is something of a hodge podge. I include a book from Peter Kropotkin, even though it is much older than the remaining non-fiction works, since I am currently one third, maybe, through it.

• Novels (Ken Macleod, in The Cassini Division has a more complete list as chapter titles.)
• Edward Bellamy. 1888. Looking Backward.
• Ursula K. Leguin. 1974 The Dispossessed.
• William Morris. 1890. News from Nowhere.
• Charlotte Perkins Gilman. 1915. Herland. Apparently this is one of a genre of feminist utopias.
• Francis Spufford. 2010. Red Plenty.
• Analyses and detailed proposals.
• Michael Albert. 2003. Parecon: Life After Capitalism.
• Rutger Bregman. 2017. Utopia for Realists: The Case for a Universal Basic Income.
• Gerald Cohen. 2009. Why Not Socialism?
• Theodore Burczak. 2006. Socialism after Hayek.
• Paul Cockshott and Allin Cottrell. 1993. Towards a New Socialism.
• David Ellerman. 2021. Neo-Abolitionism: Abolishing Human Rentals in Favor of Workplace Democracy.
• Geoffrey Hodgson. 2019. Is Socialism Feasible? Torwards an Alternative Future.
• Bruno Jossa. 2020. Managing the Cooperative Enterprise: The Rise of Worker-Controlled Firms.
• Janos Kornai. 1992. The Socialist System: The Political Economy of Communism.
• Peter Kropotkin. 1892. The Conquest of Bread.
• Guinevere Liberty Nell. 2010. Rediscovering Fire: Basic Economic Lessons from the Soviet Experiment to Eliminate the Market.
• Alex Nove. 1983. The Economics of Feasible Socialism.
• David Schweickart. 2002. After Capitalism.
• Joseph Stiglitz. 1996. Whither Socialism.
• Philippe Van Parijs. 2019. Basic Income: A Radical Proposal for a Free Society and a Sane Economy.

There is no scarcity of informed ideas on how a better society might function.

A Letter From Marx To Engels In 1867 On The First Volume Of Capital

This is another letter in a series I have been transcribing in which Marx discusses Capital. In this letter, he says one of the two best points in his book is his discussion of labor expressed in use value or in exchange value. Since I have not read (an english edition of) the first edition, I cannot be sure of my ground here. Apparently, Marx revised Chapter 1 quite extensively among editions. Anyways, I think this expression of labor gets at the distinction between concrete and abstract labor activities. I think the distinction between labor power and labor is a different, albeit related, distinction. Marx says that the other best point is his treatment of surplus value as an abstraction of profit, interest, and rent. I think this is an innovation not to be found in Ricardo or any other of his predecessors.

I wonder if Marx would have lived long enough to complete volumes 2 and 3, whether he would have had more concrete material and extracts from blue books in the later volumes, since Volume 3, at least, is at a lower level of abstraction. Given his work habits, he would have had to live a very long time. Every time he said he was going to write a few sentences on political economy, he wrote paragraphs. When he tried to write a paragraph, he wrote chapters. And a chapter would become a book. And all needed to be revised.

24 August 1867

Dear Fred,

I have received no further corrected proofs since the 2 last that I sent you. I am exceedingly vexed with Meissner. He has obviously held back what Wigand has sent him in order to send everything at once - and save 4d. postage!

The same Meissner wrote me last week that he is printing a certain part of my preface specially (and he has indeed made the right choice) to send to the German newspapers. I wrote asking him to send me COPIES of it at once. I reckoned that you would translate the thing into English (I shall then give it to The Bee-Hive, which is taken by Mill, Beesly, Harrison, etc.), and Lafargue with Laura's help into French for the Courrier français, finally I wanted to send ONE COPY to my correspondent in America. To save the 4d., Meissner has sent nothing. He will be sending it all together. But a great deal of time is lost in the process!

The best points in my book are: 1. (this is fundamental to all understanding of the FACTS) the two-fold character of labour according to whether it is expressed in use-value or exchange-value, which is brought out in the very First Chapter; 2. the treatment of surplus-value regardless of its particular forms as profit, interest, ground rent, etc. This will be made clear in the second volume especially. The treatment of the particular forms in classical political economy, where they are for ever being jumbled up together with the general form, is an olla potrida.

Please enter your desiderata, critical remarks, QUERIES, etc., on the corrected proofs. This is very important for me, as I am reckoning on a 2nd edition sooner or later. As regards CHAPTER IV, it was a hard job finding things themselves, i.e., their interconnection But with that once behind me, along came one BLUE BOOK after another just as I was composing the final version, and I was delighted to find my theoretical conclusions fully confirmed by the FACTS. Finally, it was written to the accompaniment of CARBUNCLES and daily dunning by creditors!

For the conclusion to the 2nd book (Process of Circulation), which I am writing now, I am again obliged to seek your advice on o ne point, as I did many years ago.

Fixed capital only has to be replaced in natura after, say, 10 years. In the meantime, its value returns partially and gradatim, as the goods that it has produced are sold. This PROGRESSIVE RETURN of the fixed capital is only required for its replacement (aside from REPAIRS and the like) when it becomes defunct in its material form, e.g., as a machine. Prior to that, however, these SUCCESSIVE RETURNS are in the capitalist's possession.

Many years ago I wrote to you that it seemed to me that in this manner an accumulation fund was being built up, since in the intervening period the capitalist was of course using the returned money, before replacing the capital fixe with it. You disagreed with this SOMEWHAT SUPERFICIALLV in a letter. I later found that MacCulloch describes this SINKING FUND as an accumulation fund. Being convinced that no idea of MacCulloch's could ever be right, I let the matter drop. His apologetic purpose here has already been refuted by the Malthusians, but they, too, admit the FACT.

Now, as a manufacturer, you must know what you do with the RETURNS on capital fixe before the time it has to be replaced in natura. And you must answer this point for me (without theorising, in purely practical terms).

Salut

Your

K. M.

Salut to Mrs Lizzy!

The children are still at Royan, near Bordeaux.

The Emergence Of Non-Monotonic Variations In The Economic Life Of A Machine

Figure 1: A Part of the Parameter Space

1.0 Introduction

This post presents a perturbation of an example from Salvatore Baldone. It follows the style of some posts that almost add up to a draft research paper.

A widespread view among Austrian-school and mainstream economists is mistaken. Given competitive markets, if the supply of capital were increased, in some sense, the rate of profits would supposedly be driven down. At the level of abstraction here, no distinction exists between the interest rate and the rate of profits. A lower rate of profits would incentivize capitalists to adopt more capital-intensive techniques and thereby increase the output produced per worker. A more capital-intensive technique is supposed to be associated with an increased period of production. Some such regularity seems to me to be necessary to make sense out of Austrian Business Cycle Theory.

This post presents a perturbation of an example in which net output conisists of a single commodity. Capital inputs are heterogeneous. One input consists of a machine. Sometimes truncating the economic life of the machine results in a more capital-intensive technique of production. This post emphasizes the disconnection between capital-intensity and physical properties of how long machinery is operated by capitalists maximizing their profits.

2.0 Technology

Table 1 presents coefficients of production in a perturbation of an example from Baldone (1974). With the first process, workers, under the direction of mangers of firms, manufacture new machines. The remaining three processes are used to produce corn. The second process requires an input of a new machine, as well as seed corn. A one-year old machine is jointly produced with the output of corn. The third process jointly produces an output of a two-year old machine with corn. The fourth process exhausts the physical life of the machine. Only corn is produced as a output in this process.

Table 1: Technology

Input	Industry	Agriculture
	I	II	III	IV
Labor	a0,1	39/200	117/200	39/100
Corn	a1,1	39/100	141/250	117/200
New Machines	0	1	0	0
1-Year Old Machines	0	0	1	0
2-Year Old Machines	0	0	0	1
Output
Corn	0	1	1	1
New Machines	1	0	0	0
1-Year Old Machines	0	1	0	0
2-Year Old Machines	0	0	1	0

Corn is both a consumption good and a capital good, insofar as it is an input into all four processes. Technology is not yet fully specified in this example, with numerical values undefined for two coefficients of production. I assume free disposal of old machines. The choice of technique here is equivalent to the choice of the economic life of the machine. Table 2 lists the processes operated for the available techniques, each of which can produce a net output of one bushel corn. Under Alpha, the machine is junked after operating one year. Under Beta, it is junked after two years. Under Gamma, it is operated for its full physical life.

Table 2: The Technique of Production

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

3.0 Prices of Production and the Choice of Technique

The economic life of the machine is chosen on the basis of cost-minimization. Accordingly, I consider prices of production for each technique. A bushel corn is numeraire. I assume that wages are paid out of the surplus product at the end of the year. Each operated process yields the same rate of profits.

These assumptions are enough to derive, for each technique, wages and prices as functions of the rate of profits. Three equivalent methods can be used to find the cost-minimizing technique at any given rate of profits:

• Construct the wage frontier as the outer envelope of the wage curves for the technique. This is a pure fixed capital example.
• Look for negative prices for old machines. Beta cannot be cost minimizing if a one-year old machine has a negative price at prices of production for the Beta technique. Gamma cannot be cost-minimizing if a two-year old machine has a negative price at Gamma prices.
• Determine if extra profits can be made by adopting a process not operated in calculating prices of production. Does operating the machine for two years yield extra profits at Alpha prices? For three years at Beta prices?

For the pure fixed capital model, these three methods of analyzing the choice of technique are equivalent. In the general case of joint production, the outer envelope of the wage curves does not always correspond to the cost-minimizing technique.

4.0 Two Fluke Points and the Perturbation of Coefficients of Production

Figure 2 illustrates the analyis of the choice of technique for the specified values of the direct labor input and corn input, per machine produced in the industrial process. A switch point between Beta and Gamma exists at approximately 61 percent. Around this switch point, a lower rate of profits is associated with an extension of the economic lifetime of the machine and an increased output per worker. Score one for Böhm Bawerk. These parameters have been carefully chosen to yield two fluke switch points with Alpha, one at the maximum wage and one at the maximum rate of profits. Aside from these flukes, this example seems straightforward.

Figure 2: An Example Of Two Fluke Switch Points

I now consider local perturbations of these two coefficients of production. Figure 1, at the top of this post, depicts the part of the parameter space under consideration. The machine varies in efficiency as it is used in agriculture, and these perturbations leave unchanged the needed inputs of labor and corn to operate with the machine at each age. The parameters partition the parameter space around the fluke case in Figure 2. The wage frontier does not differ qualitatively in each of these four regions, but it does differ among regions.

The wage frontier to the northeast looks like the frontier in Figure 2, but the switch points on the axis for the wage or the rate of profits no longer appear. As noted, the single switch point between Beta and Gamma conforms to outdated Austrian and marginalist intuition.

5.0 Non-Monotonic Economic Lifetime of a Machine to the Northwest

The region to the northwest, as illustrated in Figure 3, is more interesting. This is not a reswitching example. Nor is it an example of capital-reversing in this region of the parameter space. Prices can be used to evaluate the payments made for advances for capital inputs for each technique. Around the two switch points, a lower rate of profits is associated with a larger value of capital per unit output for the newly cost-minimizing technique. More capital per worker is used, and output per worker is increased. With a lower rate of profits, the machine is operated two years, three years, and then one year. At the first switch point, a shorter economic life of a machine is associated with greater capital-intensity. This result seems to pose some difficulty for traditional formulations of Austrian capital theory.

Figure 3: An Example Of Non-Monotonic Variations in the Economic Life of a Machine

6.0 Capital-Reversing to the Southeast

Figure 4 illustrates the region in the parameter space to the southeast. This is an example of capital-reversing but not of the reswitching of techniques. The wage curves for the Alpha and Beta techniques intersect twice, with the first intersection within the wage frontier. Around the second switch point, a lower rate of profits is associated with a smaller value of capital per unit output. Less capital per worker is used, and output per worker is lower at a lower rate of profits. Long run marginalist theory is falsified. With a lower rate of profits, the machine is operated one year, two years, and then three years. At the second switch point, a longer economic life of a machine is associated with a decrease in capital-intensity. Austrian capital theory is not validated here, either.

Figure 4: An Example Of Capital-Reversing

7.0 Reswitching to the Southwest

In the southwest, both these difficulties for Austrian capital theory arise, as illustrated in Figure 5. This is now a reswitching example. At a low or high rate of profits, the machine is operated for one year. The interval of the rate of profits between these two intervals in which Alpha is cost-minimized are partitioned by another switch point. Gamma is cost-minimizing in one partition, and Beta is cost-minimizing in the other. At the first switch point, a shorter economic life of the machine is associated with a lower rate of profits and the adoption of a more capital-intensive technique. At the third switch point, a longer economic life of a machine is associated with a smaller rate of profits and a decrease in capital-intensity.

Figure 5: A Reswitching Example

8.0 Conclusion

Harwick (2022) has noted that some followers of the Austrian school have recently tried to consider Austrian capital theory separately from business-cycle theory. Lewin and Cachanosky (2019) consider a financial measure of capital-intensity, namely the average duration of an investment project. Around any switch point, an increased Duration is (tautologically?) associated with a lower interest rate. As noted by Fratini (2019), an increased capital intensity, in this sense, is associated with reduced net output per worker around a so-called 'perverse' switch point. Noting that a technique exhibiting joint production cannot always be reduced to an infinite series of dated labor inputs, I do not know how to calculate Duration for the above example. (Those wanting to address this implicit challenge might want to start with this example to warmup.) But even so, an increased Duration can be associated with the reduction in the economic life of a machine.

Anyways, a reduction in the economic life of a machine is consistent with an increase in capital-intensity. Capital-intensity is assessed above by evaluating the price of inputs, either for a given net output or per worker. This association between a shorter economic life and greater capital-intensity can arise around a switch point in which a smaller rate of profits incentives the adoption of a more capital-intensive technique, with a consequent greater net output per worker. It can also arise around a ‘perverse’ switch point in which a less capital-intensive technique is adopted at a lower rate of profits. Neither type of switch point is a fluke case, as can be seen by contrasting such switch points with genuine fluke cases.

Reference

• Salvatore Baldone. 1974. Il capitale fisso nello schema teorico di Piero Sraffa. Studi Economici XXIV(1): 45-106. Translated in Pasinetti (1980).
• Saverio M. Fratini. 2019. A note on re-switching, the average period of production and the Austrian business-cycle theory. Review of Austrian Economics 32: 363-374.
• Cameron Harwick. 2022. Unmixing the metaphors of Austrian capital theory. Review of Austrian Economics 35: 163-176.
• Peter Lewin and Nicholas Cachanosky. 2019. Austrian Capital Theory: A Modern Survey of the Essentials. Cambridge: Cambridge University Press.