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| Figure 1: A Partition of Part of the Parameter Space |
This post is a continuation of the example in a previous post. That example is of the recurrence of truncation without reswitching. I here consider perturbations of selected coefficients of production and of relative markups in the two industries.
2.0 Perturbation of Efficiency of Circulating Capital for Old MachinesThe first industry in the example produces machines, and the second produces corn. Machines constitute fixed capital, and corn functions as circulating capital, as well as a consumption good.
Two processes are available in each industry. The second process uses an old machine to produce the output of that industry, whether a new machine or corn. Old machines are specific to the industry in which they were (jointly) produced. The choice of technique, in this model of pure fixed capital, is equivalent to choosing to truncate the economic life of a machine in either industry. The four possible techniques are defined, along with the technology, in the previous post.
Consider perturbations of a1,2 and a1,4. The first coefficient of production is the amount of corn input, as circulating capital, needed to operate an old machine in the machine industry. The second coefficient of production is the input of corn needed to operate an old machine in the corn industry.
Figure 1, at the top of this post depicts a partition of the resulting parameter space. The variation of the choice of technique with distribution is invariant in each numbered region. Table 1 lists the cost-minimizing techniques, in order of an increasing rate of profits in each region.
| 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. |
The diagram yields the following results:
All these regions are around a quintuple fluke switch point. The partition between regions 1 and 5 occurs for parameters for which managers of firms are indifferent, when the wage is zero, about the economic life of a machine in producing new machines. The partition between regions 1 and 2 occurs when managers of firms are indifferent, also at a wage of zero, about the economic life of a machine in producing corn. The intersection of these two partitions must also be an intersection of the other three partitions.
3.0 Perturbation of Relative MarkupsNow suppose the technology is fixed, as in the post. Let s1 r be the rate of profits in the machine industry, and s2 r the rate of profits in the corn industry. As a normalization condition, I assume the sum of the relative markups is unity:
s1 + s2 = 1
Figure 2 displays the effects on the choice of technique of perturbations of persistent relative markups.
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| Figure 2: Effects on The Choice of Technique of Perturbations of Relative Markups |
High enough market power for the corn industry, as compared to market power in the machine industry, can eliminate the possibility of extending the economic life of the the machine in the corn industry. This is seen in region 5, to the left on the graph. Persistent high enough market power for the machine industry, can also eliminate the recurrence of the truncation of machines in the corn industry. This is seen in region 3 to the right on the above graph.
4.0 ConclusionThe choice of technique varies with variations in distribution between wages and profits. Both technical progress and changes in market power can have similar effects, in the large. In this example, both can bring about or eliminate the recurrence of the truncation of the economic life of the machine in one industry. Technical progress, however, has the benefit of increasing productivity.
Steve Keen appeared on the 30 April epsiode of 1Dime Radio, a podcast. Towards the start, he tells a story:
"My change actually came from a very technical piece of economics because my first-year lecturer is a still a good friend these days, Professor Frank Stillwell - back in those days, he was Doctor Frank Stillwell – explained what's called the theory of the second-best in the first-year lectures. And this is something you normally learn in a third or fourth-year honors course or master's or PhD qualifying. And by the time you’ve got to that level, most people were being so saturated with neoclassical thinking that they would have just regarded it as, 'Oh, that’s a nice little curiosity', and they forget about it. What it showed was that = I think it got the originators the Nobel prize at one point – they show that if you were two steps away from what's regarded as perfection by neoclassical economics, then moving one towards it, and not the whole two, would actually make social welfare worse. And the example that Frank used was if you have wage negotiations, the ideal according to neoclassical theory is that you have workers who bargain on their own personnel right and firms who, again, bargain for employees on an individual basis – no collusion within labour or capital.
But the real world is you’ve got trade unions on one side and employer associations on the other. So in the neoclassical view, you get equilibrium where the worker gets paid their marginal product. That's the ideal. When you allow that there's both trade unions and monopolies (employer associations), you move to another point where it's indeterminate what the wage will be and it’s a bargaining point which might make the firms better off or the workers better off compared to the ideal. But if you abolish one or the other, either get rid of the trade union or get rid of the employer association, the outcome is necessarily lower social welfare than the previous case where had both the trade union and the employer association.
And I fell for the conventional argument. I accepted all the idea of supply and demand and equilibrium and so on. And then to have if pointed out that if you take into account the reality there’s plenty of distortions from what's called the perfectly competitive ideal, then getting rid of them sequentially will make things worse. I thought, there's got to be something wrong with the theory if you can simply demolish it so easily. So I checked my textbook. There was no mention of the theory of the second-best there
I then went down to the economics department library, which is in the same building as where the lectures were, the Merewether Building at Sydney University. And I went looking for the journal papers. I found the original. And then I was horrified that this is not covered in the textbooks.
So I went to the journals again looking for the most recent journal papers. And I found one by Paul Samuelson which was called – first of all I found a journal paper by a Marxist and that surprised me. That was in the Cambridge Journal of Economics by Bhaduri. And I was amazed that a Marxist got into a journal. That surprised me. But then I read Samuelson, a paper called 'A summing up'. And he basically conceded defeat in a debate over the definition of capital which I did even know was happening. But it was actually taking place between 1960, when Piero Sraffa published A Production of Commodities by Means of Commodities, through to the – probably petered out in the late sixties. No mention of it, Samuelson conceded defeat in that paper, but you read his textbook which I had at the time – no recognition of the dispute there either. So I thought I'm being lied to by my textbooks.
And I stopped reading the textbooks. I read them anyway for reference, obviously. But I go and take a look at the journal papers and seeing what's being said in the journals. And the gap between what I was being taught versus the journals wasn't a case that I was getting the simplified version in the textbooks and the sophisticated stuff is in the journals. I was seeing completely contrary results for absolutely fundamental arguments in the textbooks. And I just thought these textbooks are mendacious. Whether they know it or not, they're lying about the nature of economics.
So that was my breakaway point and I’ve never looked back. So that's why I regard economics as unscientific in the extreme, because there have been so many anomalies and so many logical disproofs, and so many empirical failings, this theory should not even be around anymore. It should be like phlogiston in chemistry. But it still dominates economics today. And they are so bloody arrogant about it. That's the other terrifying thing. They are so sure that they’ve got the right answers to everything when history and logical analysis shows that they've got the wrong answers to everything." -- Steve Keen (My transcription)
Richard Lipsey and Kelvin Lancaster published 'The general thory of second best' in 1956-1957, in the Review of Economic Studies. Neither won the Nobel prize. Lancaster won the John Bates Clark medal, which is very prestigious.
Keen does not go into this, but I think a distinction exists between the theory of the second best and the results of the Cambridge capital controversy (CCC). John Eatwell is good on this disticntion. The theory of the second best is one of a number of imperfections and frictions, like transaction costs, information asymmetries, principal agent problems, externalities, search costs, and incomplete contracts. Underlying these imperfections and frictions is an ideal theory. But the CCC shows that this ideal theory is incoherent. Maybe I am too firm on this distinction. If you are clear on all these imperfections and frictions, you know that the ideal is unattainable anyways, whatever policy the government adopts. Talk of government non-intervention in the economy is incoherent.
I think Keen is conflating Amrit Bhaduri's 1969 Economic Journal article, 'On the significance of recent controversies on capital theory: a Marxian view' with later articles.
Paul Samuelson did modify the tenth edition of his textbook. But later editions are befuddling.
I might as well say something about how I developed my views, keeping in mind that I have never been an academic economist. Sometime in the 1980s, I came across a reference to Robinson as the 'British Galbraith'. I had always like Galbraith, who I thought of as a popular writer and advocate of liberalism. So I looked up Joan Robinson's writing. I came across much about Sraffa and the Cambridge capital controversy. I ended up reading some of the same journal articles as Keen. I do not know that you should trust my self-depiction, but perhaps I continued looking for some, any response that defends what is in the textbooks for intermediate microeconomics. I could see that the question was not merely whether aggregate production functions, in macroeconomics, are a useful simplification.
Those defending marginalist economics in the CCC do not end up supporting the view in the textbooks. Capital is not a factor of production. If follows that interest is not a payment for the services of a factor of production. The aggregate production function is theoretically unfounded. Marginal productivity is not a theory of distribution. Equations relating payments to marginal products are merely part of the formulation of a system of general equilibrium. No theoretical foundation exists for well-behaved supply and demand functions in, say, labor markets. Maybe I am wrong, but I see it as very difficult to defend mainstream academic teaching as well-informed and honest.
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| Figure 1: Wage Curves In The Example |
I have presented this example before. This example is another case of exploring or demonstrating code written for Matlab or Octave.
The structure of the example is the minimum multi-industry example with circulating and fixed capital in all industries and in which the choice of technique is to select the economic life of a machine.
The recurrence of truncation is like the recurrence of a process in single production. As far as I know, no numeric example exists in the literature of the recurrence of truncation without reswitching. This example might have been surprising if I were writing half a century ago. Its possibility is obvious in the work of Bertram Shefold, Heinz Kurz & Neri Salvadori, Ian Steedman, and others. Although reswitching and capital-reversing do not arise in the example, the reverse substitution of labor does.
2.0 Technology and TechniquesTwo industries exist in the example. One industry produces machines, and the other industry produces corn. Corn is a consumption good, the good for circulating capital, and the numeraire. Machines are fixed capital. Each machine has a physical life of two years. Old machines cannot be transferred between industries. I assume constant returns to scale (CRS) and the free disposal of old machines. Labor is advanced and paid out of the surplus of corn.
Tables 1 and 2 show the inputs and outputs for each process known to the managers of firms. For example, the inputs, at a unit level of operation, consist of 1/10 person-years, 1/16 bushels corn, and one new machine. The outputs, available after a year, are two new machines and one machine a year older.
| Input | Industry | |||
| Machine | Corn | |||
| I | II | III | IV | |
| Labor | 1/10 | 8 | 43/40 | 1 |
| Corn | 1/16 | 3/20 | 1/8 | 53/200 |
| 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 |
| 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 |
The machines operate an non-constant efficiency in both industries. An old machine, in the machine industry, is used to produce more new machines than a new machine. The inputs of labor services and corn increase with the age of the machine. In the corn industry, an ole machine is used to produce less corn than a new machine. The input of labor services decrease and the corn input increases with the age of the machine.
With this specification of the technology, the economic life of the machine must be chosen in each industry. Table 3 lists the available techniques. The machine is truncated in both industries in the Alpha technique. The machine is operated for its full physical life in both industries in the Delta technique. In Beta and Gamma, the machine is truncated in one industry and operated for its full physical life in the other.
| Technique | Processes |
| Alpha | I, III |
| Beta | I, II, III |
| Gamma | I, III, IV |
| Delta | I, II, III, IV |
The economic life of a machine is chosen to minimize cost. A system of equations for prices is associated with each technique. This system can be solved. In the solution, the wage is a function of the rate of profit. Each price of a produced commodity is also a function of the rate of profits.
Figure 1 shows the wage curves, for the four techniques in the example. The cost-minimizing technique at each wage or rate of profits is the technique with its wage curve on the outer frontier. The cost-minimizing techniques are indicated on the figure. Maybe I should experiment with perturbing parameters to see if I can get a more visually obvious graph. Figure 2 shows an enlargement, emphasizing rates of profits around the switch point between Gamma and Delta.
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| Figure 2: Wage Curves In The Example (Enlarged) |
At any rate, the cost-minimizing techniques, in order of an increasing rate of profits, are Alpha, Gamma, Delta, and Beta. Each pair of techniques at a switch point on the frontier differs in one process. A switch point in which the economic life of a machine differs in both industries would be a fluke case. No fluke switch points exist in this example, without perturbing some coefficients of production.
4.0 Prices of Old MachinesIdentifying when prices of old machines are negative provides another method of analyzing the choice of technique in models of pure fixed capital. A negative price indicates that the economic life of a machine should be shortened. The machine should be truncated and discarded.
Figure 3 plots the price of old machines in the machine industry, for the two techniques in which old machines are operated in this industry. The switch points, at which the price of an old machine is zero, are indicated. As can be seen in Figure 2, the switch point between Alpha and Beta is not on the outer frontier.
For rates of profits less than that at the switch point between Gamma and Delta, the price of an old machine in the machine industry is negative for the Delta price system. If the Delta technique were in operation, prices would signal that machines in the Delta industry should be truncated. This trunction results in the Gamma tecnique being adopted.
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| Figure 3: The Price of an Old Machine in Machine Production |
Figure 4 plots the price of old machines in the corn industry. Old machines are operated in this industry only for Beta and Delta. Since the price of these old machines are negative, in the Gamma price system, for rates of profits less than the rate at which the price is zero, the machine is truncated at these rates and the Alpha technique is adopted. Likewise, at rates of profits greater than the rate at which the price of this machine is zero, in the Delta system, the machine is truncated and the Beta technique is cost-minimizing at these rates.
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| Figure 4: The Price of an Old Machine in Corn Production |
This analysis of prices of old machines has re-justified the analysis of the choice of technique in Section 4.
5.0 Extra Profits in Extending the Economic Life of MachinesA third method of examining the choice of technique is available.
Under Alpha and Gamma, the machine is truncated in the machine industry. The price of an old machine in the machine industry is zero under those price systems. Figure 5 shows extra profits, for each technique, available in operating the machine for a second year. if the life of this type of machine is extended under Gamma, the Delta technique is adopted. Extra profits are available in so extending the life of the machine at any rate of profits greater than at the switch point between Gamma and Delta. Gamma cannot be cost-minimizing in this range.
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| Figure 5: Extra Profits in the Machine Industry |
The machine is truncated in the corn industry for Alpha and Beta. Figure 6 shows extra profits in the corn-industry, for all techniques, in operating the machine for a second year. Extra profits cannot be obtained for Alpha up to the switch point between Alpha and Gamma. Likewise, extra profits are not available for Beta, in extending the life of the machine in corn-production, for rates of profits greater than at the switch point between Beta and Delta. This method of analyzing the choice of technique, not surprisingly, yields the same result as the other two.
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| Figure 6: Extra Profits in the Machine Industry |
6.0 Recap
The above has illustrated three equivalent methods of analyzing the choice of technique for a pure fixed capital model. Table 4 summarizes the results for this numerical example. The bounds on the ranges of the rates of profits are approximate. Matlab has a funtion, roots(), that returns the (possibly complex) zeros for a polynomial of any degree. I use this function in finding the intersections of wage curves in this example.
| Range | Technique | Truncation |
| 0 ≤ r ≤ 70.21% | Alpha | Machines truncated in both industries. |
| 70.21% ≤ r ≤ 71.19% | Gamma | Machines truncated in machine-production. |
| 71.19% ≤ r ≤ 87.5% | Delta | Machines operated at full physical life in both industries. |
| 87.5% ≤ r ≤ 122.8% | Beta | Machines truncated in corn-production. |
At any rate, the machine is truncated in corn-production when both the Alpha and the Beta technique are cost-minimizing. The truncation of the machine in corn-production recurs, being part of the cost-minimizing technique at extremes of low and high rates of profits. This is not, however, an example of the reswitching of techniques.
Negative real Wicksell effects occur at all four switch points. Around each switch point, a lower rate of profits and higher wage is associated with a greater net output of corn per person-year. At the switch point between Alpha and Gamma, truncation in the corn industry is a switch to a more capital-intensive technique. Likewise, at the switch point Gamma and Delta, truncation in the machine industry is a switch to a more capital-intensive technique. As usual, these results disagree with Austrian capital theory and the ideas of economists of this school about roundaboutness.
Around the switch point between Alpha and Gamma, a lower rate of profits or higher wage is associated with truncation in the corn industry and a greater gross output of corn per person-year hired in the corn industry. Around the switch point between Delta and Beta, contrawise, a lower rate of profits or higher wage is associated with the extension of the economic life of the machine in the corn industry and a decrease in the gross output of corn per person-year hired in the corn industry. This second switch point is a manifestation of the reverse substitution of labor, one of those 'perverse' phenomena found in the Cambridge capital controversy.
Carl Menger has a theory of consumer demand, in his Principles of Economics. This theory, one expression of utility theory, is ejected or ignored by other marginalist economists. Bohm-Bawerk is an exception. He also puts forth this theory. For those who want to read something shorter, I recommend William Smart's 1891 An Introduction to the Theory of Value. Heinz Kurz has recently written about Menger.
I take current theory to be revealed preference theory, which was developed by Paul A. Samuelson. Gerard Debreu's 1959 Theory of Value: An Axiomatic Analysis of Economic Equilibrium is canonical. in the theory, each consumer has a preference relation over a space of goods. Suppose all goods can be enumerated. Debreu has No. 2 Red Winter Wheat as an example of one good. Suppose a consumer is presented with vectors of n goods, where n is the number of goods available. Each vector specifies the quantity of each good available. The consumer is assumed to be able to tell, for each pair of vectors, whether they prefer the first to the second, they prefer the second to the first, or they are indifferent between them. Given certain assumptions on preferences, a utility can be assigned to each vector. This utility has some of the properties of numbers. You may not have the mathematics to understand some expositions of this theory, and other expositions exist, for example, in terms of choice functions.
Menger, by contrast, looks at one good at a time. He has a couple of chapters on the theory of the good. In his chapter on value, he classifies wants or needs into different classes. For example, food might be a class. A good, say, water, might go into several classes. You can drink water, use it to water your lawn, or use it to fill a swimming pool. These might be three different classes. The consumer has ranks, in each class, of satisfactions or utilities. The first gallon of water, in the drinking class, might have a rank of 10, while each successive gallon has a lower rank. When the consumer obtains a new gallon of water, they must look at the next satisfaction to be obtained, with the given distribution of existing goods among the classes. The consumer will then allocate this next gallon among these uses accordingly.
None of the structure in Menger's theory survives in modern economics. I think even Kelvin Lancaster's1966 New approach to consumer theory is something different.
Other aspects of Menger’s book are also obsolete. But I want to only focus on one aspect at a time.
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| Figure 1: Wage Curves For A Technique In The Example |
I have explored this example from Baldone before, including perturbations of coefficients of production. My purpose here is to demonstrate that my Matlab code for Sraffian analysis can yield the correct results. (I have an off-by-one error that I hard-coded around in obtaining these graphs.)
My favorite method of analyzing the choice of technique applies to models of pure fixed capital. In such models, machines that last over multiple production periods are the only element of joint production. If a machine does not have constant efficiency over its physical life, the analysis of the choice of technique includes a decision on the economic life of the machine. The choice of technique can still be analyzed by the construction of the wage frontier as the outer envelope of wage curves. Unlike in single production, a wage curve can slope up off the frontier.
Baldone's numerical example illustrates an equivalent method for analyzing the economic life of a machine. It focuses attention on negative prices of old machines. The cost-minimizing technique is such that old machines are discarded, not operated. And it is an example of the reswitching of techniques.
2.0 Technology, Techniques, and Quantity FlowsEach column in Tables 1 and 2 defines a production process. Managers of firms know about each process. The first produces new machines, and the remaining three produce corn with machines of various vintages. For instance, a bushel corn and a one-year old machine are produced, in the second process, from inputs of 1/5 person-years of labor, 2/5 bushels corn, and one new machine.
| Input | Process | |||
| (I) | (II) | (III) | (IV) | |
| Labor | 2/5 | 1/5 | 3/5 | 2/5 |
| Corn | 1/10 | 2/5 | 289/500 | 3/5 |
| New Machines | 0 | 1 | 0 | 0 |
| One-Year Old Machines | 0 | 0 | 1 | 0 |
| Two-Year Old Machines | 0 | 0 | 0 | 1 |
| Output | Process | |||
| (I) | (II) | (III) | (IV) | |
| Corn | 0 | 1 | 1 | 1 |
| New Machines | 1 | 0 | 0 | 0 |
| One-Year Old Machines | 0 | 1 | 0 | 0 |
| Two-Year Old Machines | 0 | 0 | 1 | 0 |
I call Alpha the technique in which the machine is disposed of after one year and Beta the technique in which the machine is discarded after two years. In Gamma, the machine is run for its full three physical years
Suppose Alpha is adopted, and the first two processes are operated at a unit level. A new machine is simultaneously produced by the first process and operated to its economic life in the second. One bushel corn is produced. One half bushel is used to replace the corn input, leaving a net output of 1/2 bushel corn. This net output is produced by 3/5 person-years labor. Thus, Alpha requires 1.2 person-years per net bushel output ( = (3/5)/(1/2) = 6/5). I leave it for the reader that Gamma requires approximately 1.2103 person-years per net bushel corn, and that Beta requires approximately 1.3015 person-years per net-bushel produced.
3.0 PricesIn a vertically integrated firm, new and old machines are not sold on markets. Nevertheless, the accountants must enter prices on the books. The accounting I outline here can be used to derive the formula for an annuity if the efficiency of the machine were constant. However, since that is not the case, a general approach to depreciation is illustrated.
Let r be the interest rate, as given from the market, w the wage, p0 the price of a new machine, p1 the price of a one-year old machine, and p2 the price of a two-year old machine. The interest rate is also known as the rate of profits. When the Gamma technique is operated, prices must satisfy the following system of four equations:
(1/10)(1 + r) + (2/5) w = p0
((2/5) + p0)(1 + r) + (1/5) w = 1 + p1
((289/500) + p1)(1 + r) + (3/5) w = 1 + p2
((3/5) + p2)(1 + r) + (2/5) w = 1
I take the wage as paid at the end of the year, and all prices are expressed in terms of the net product.
If the interest rate is given, the above system consists of four linear equations in four variables. It can be solved.
The price systems for the other two techniques are a subset of those. The price system for Beta, for instance, consists of the first three equations, with the price of a two-year old machine set to zero.
4.0 Non-Negative Prices and the Choice of Technique"With decreasing or changing efficiency ... a problem of the choice of technique, that is, of the optimal truncation date, arises. Premature truncation is advantageous as soon as the price (book value) of a partly worn out instrument of production becomes negative. Since the price of a machine (either new or 'aged') is equal to the capital value one gets by discounting all future net recipts that may be obtained by further use of it, where the going rate of profit is taken as the discount rate, negative prices would indicte 'losses' and would thus contradict the assumption of a fully settled competitive position of the economy." -- Kurz and Salvadori (1995: 212).
I can find when the price of each machine is positive. For new machines (Figure 2), their prices are positive:
The upper limits are approximate. The wage curves in Figure 1, at the top of this post, intersect the axis for the rate of profits at these upper limits.
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| Figure 2: Prices of New Machines |
One-year old machines have positive prices (Figure 3):
Under Alpha, the machine is discarded after one year, and the prices of old machines are identically zero. Beta is not operated outside the limits in which the price curve for Beta intersects the abscissa in Figure 3. If the machine were being truncated after two years, it would pay to discard it after one year. The same applies to Gamma. The analysis, so far, shows that Alpha would be adopted at the extremes of low and high rates of profits,
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| Figure 3: Prices of One-Year Old Machines |
Two-year old machines have positive prices (Figure 4):
Since the price of a two year old machine is negative for rates of profits greater than at the switch point, Gamma will not be operated at those rates of profits.
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| Figure 4: Prices of Two-Year Old Machines |
I can now summarize the analysis of the choice of technique for this example. Managers of firms will not adopt a technique when the outputs of a process in the technique has a negative price. Thus, each technique will be adopted in the following intervals:
Now, I can look at what happens around the three switch points:
Only the middle switch point validates Austrian capital theory. Clearly, economists of the Austrian school have made mistakes in logic.
I like to note that the above argument is not about aggregation.
5.0 ConclusionThe above constitutes a proof that Austrian capital theory is mistaken. It relies on an identification, in the example, of more roundaboutness with a longer economic life of a machine. Austrian economists have tried to express their central insight that a greater use of capital is equivalent to a greater use of time in several disparate ways.
Perhaps greater roundaboutness should be identified with the use of different, better machines. By putting aside some time each day, Crusoe can make a net, instead of relying on whatever lies about at hand when catching fish. Or perhaps roundaboutness should be measured by a average period of production. Or by a financial measure of duration. What about those Hayekian triangles?
Since the central insight happens to be wrong, each of these formulations can be demonstrated to be, at best, ad hoc. But for each formulation, to be shown wrong in detail, requires a separate argument. Such can be provided and has been provided for most. Both Austrians and more mainstream marginalists have been in the position, for decades, that every economist is their own capital-theorist.
ReferencesAbstract: 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. The assertion, common among some economists of the Austrian school, that more roundabout processes are more capital intensive is demonstrated to be unsustainable.
This post and these four posts make a draft paper. A draft abstraction is above. A draft of the introduction and conclusion follows.
The reswitching of techniques is probably the most surprising result from the Cambridge capital controversy. Kurz & Salvadori (1995) is a standard textbook presentation of the analysis of prices of production and of the choice of technique. Switch points, in which two techniques are both cost-minimizing at a given wage or rate of profits, are found as the zeros of certain polynomials of high degree. Reswitching occurs when two techniques have multiple switch points on the wage frontier at economically meaningful rates of profits. These zeros can be complex and, if real, need not be positive and below the maximum rate of profits. Nevertheless, no obvious rationale exists for not expecting many economically feasible switch points to exist. Then one technique will be cost-minimizing in at least two disjoint intervals of the rate of profits, if more than one switch point is on the wage frontier.
Empirical research indicates, however, that the reswitching of techniques is rare. Kurz (2020) argues that these empirical investigations, although impressive, still suffer from limitations not overcome in data collection. Only circulating capital is assumed. Heterogeneous commodities are produced in each industry, and the input coefficients vary among processes operated in an industry. Accounting conventions may assign a firm to different industries in different years, depending on the mix of products produced by each firm. Still, it is not clear why reswitching should be common, if these and other limitations in data are overcome in future work.
Schefold (2023) uses simulation to investigate the rarity of reswitching and other capital- theoretic phenomena. He randomly generates coefficients of production for alternate techniques. Wage curves are nearly affine functions. Only one, two, or maybe a few more techniques contribute their wage curves to the frontier, except near extremes for the rate of profits. The continuous variation in the cost-minimizing technique with distribution, as postulated in marginalist theory, is difficult to sustain. The reswitching of techniques does not seem likely on the wage frontier.
This article argues that reswitching can be empirically hard to observe for complementary reasons. A numerical example is created, for the corn-tractor model, that is just barely an instance of triple-switching. Fluke switch points are on the wage axis and the axis for the rate of profits. 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. A perturbation analysis partitions the parameter space with fluke switch points. The intersections of such partitions are double-fluke cases. For instance, the wage curves, with such parameters, are tangent at a switch point that is also on the wage axis. A picture of how triple-reswitching can arise emerges from an analysis of how the parameter space is divided into regions by these partitions. Technical innovation in the production of one type of tractor leads to certain trajectories through the parameter space. The emergence of triple-switching requires specific evolutions of coefficients of production. Further evolution of technology removes the possibility of triple-switching. The example also illustrates that the roundaboutness of a technique is independent of the capital-intensity of a technique.
The corn-tractor model is an extension of the Samuelson-Garegnani model. Samuelson (1962) attempts to provide a rigorous defense of aggregate marginalist theory, as in the Solow-Swan model of economic growth. Samuelson’s model consists of any number of techniques, each associated with a different type of capital good, called a ‘tractor’ here. Labor and tractors can produce a new tractor, or they produce the consumption good, called ‘corn’. Garegnani (1970), in his general treatment of an economy in which multiple commodities are produced, considers only the case of circulating capital. He shows that Samuelson’s conclusions depend decisively on the critical assumption that, for each type of tractor, coefficients of production do not vary, other than by a scale factor, between the tractor and corn industries. Steedman (2019) extends the model to a special case of fixed capital. He treats depreciation as in Sraffa’s model of joint production, instead of as radioactive decay, as in Samuelson’s approach.
An original contribution of this article is to refine the argument in Vienneau (2025b) with a more perspicacious example. It argues that coefficients of production supporting multiple switch points between two techniques can arise only temporarily, as one technique replaces another with technical progress. It also validates assertions in Steedman (2019) with numerical examples. In contrast to Samuelson (1962), double-switching can arise when each capital good is produced with the same physical capital intensity as when it is used to produce the consumption good. Triple- switching can arise when this assumption is relaxed. As an aside, the claim common among some economists of the Austrian school that more roundabout processes are more capital intensive is demonstrated to be unsustainable. This demonstration identifies a more roundabout technique with the production and use of a capital good that lasts for more time in the corn-tractor model.
The remainder of this article consists of two sections and an appendix. The next section analyzes an example in the corn-tractor model. The technology is specified for a numeric example. The system of equations for prices of production is specified and solved. A selected part of the parameter space is partitioned by fluke switch points. Switch points occurring with perturbations of coefficients of production are used to demonstrate certain aspects of capital theory. An analysis of structural economic dynamics shows how triple-switching can appear and disappear with technical progress. The final section concludes. The appendix modifies the example to partition the parameter space in a case in which double-switching, but not triple-switching, can occur.
Steedman, as in many of his papers, seems to be setting a homework problem for the advanced student:
“We therefore urge Sraffa-inspired authors to pay more attention to the analysis of fixed capital in simple models of production and hope that enough has been said here to provide a systematic basis for such further analysis.” (Steedman 2019)
This article is my answer, with the solution extended to consider perturbations of coefficients of production and a kind of structural economic dynamics. It validates the claim that triple-switching can arise in a simple example of the corn-tractor model. The physical capital-intensity varies between industries for a type of tractor that last more than one production period in this example. It also validates the possibility of double-switching, even when, for each type of tractor, the physical capital-intensity is constant across industries. This result contradicts Samuelson (1962).
The critique of Austrian roundaboutness is extended. A lower rate of profits around a switch point may be associated with the adoption of a more or a less roundabout technique. A lower rate of profits around a switch point may also be associated with more or less net output per worker. Example switch points with all four possible combinations are presented above.
An illustration is given of how parameter spaces are partitioned with fluke switch points. The resulting qualitative structure of regions is claimed to be generic. The example illustrates that in a process of technical change, with one technique replacing another, parameters corresponding to cases of multiple switch points can only be transient. The question of how prices of production relate to market prices is left unaddressed.
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