The Truncation Of The Economic Lives Of Machines

'Paradoxes' and 'Perversities'

Phenomenon	Example	Region
Reswitching	'One good'	5
	Schefold reswitching	3
	Schefold roundabout	3
	Baldone	8
Recurrence of technique (without reswitching)	Baldone	9
Recurrence of truncation (without reswitching or recurrence of technique)	Two sectors with fixed capital	2
Non-monotonic variation of economic life of machine (without reswitching or recurrence of technique or of truncation)	Baldone	10
'Non-continuous' variation in economic life of machine associated with infinitesimal variation in rate of profits	'One good'	1, 5
	Baldone	7, 8, 9, 10, 11
Increased economic life of machine associated with lower capital intensity	'One good'	1, 3, 4
	Schefold reswitching	2
	Two sectors with fixed capital	1, 2, 3, 4
	Baldone	9, 10, 11
A lower rate of profits associated with a decreased economic life of a machine	'One good'	1, 3, 4, 5
	Schefold reswitching	2, 3
	Two sectors with fixed capital	1, 2, 3, 4
	Baldone	8, 9, 10, 11
Decreased roundaboutness associated with a lower rate of profits	Schefold roundabout	2, 3, 4

I have been exploring simple models of fixed capital, of the production of commodities with machines that last more than one production period. And in these models, the efficiency of machines varies with age. An older machine might require greater care or produce more of a finished commodity after it has been broken in. The choice of technique becomes a question of the choice of the economic life of a machine. In the jargon, managers of firms decide on whether to truncate the use of machine and for how long.

One might think intuitively, but wrongly, that by first producing a machine and then using it in the production of a finished good that one was adopting a more capital-intensive technique than by directing producing the finished good. Likewise, one might wrongly believe that extending the economic life of a machine increases the capital-intensity of a technique. And that a lower rate of interest (or a higher wage) provides incentives to the managers of firms to adopt more capital-intensive techniques.

One can see that these beliefs are incorrect by looking at specific numerical examples. The table at the head of this post provides examples of curious phenomena seen for the fixed capital. Links are provided to specific examples. (The numbering of regions for the 'one good' example are not consistent over the years that I have been working on models of fixed capital.) I think that some of these effects have not been noted in the literature before, albeit I always suspect that Kurz and Salvadori's 1995 textbook might have a homework problem that I now understand the point of.

The truncation of machines is another aspect of the Cambridge Capital Controversy (CCC). But it was not made much of during the 1960s.

My research project of looking at parameter perturbations to identify fluke switch points and partitions of parameter spaces is hardly exhausted. Some research areas to investigate include:

• Create and perturb examples of reswitching and capital reversing, for example, in models of fixed capital in which machines operate with constant efficiency.
• Perturb coeficients of production and requirements for use in models with land, paying particular attention to the order of efficiency, the order of rent, extensive rent, and intensive rent.
• Perturb coefficients of production and requirements for use in general models of joint production.
• Revisit the above considering perturbations of relative markups among industries, instead of coefficients of production.
• Develop computer programs to aid in these analyses.

And besides extending my results, I still need to make an effort to submit much of what I have for publication.

I have decided that applying these results in sensitivity studies of empirical results with National Income and Product Accounts (NIPAs) is probably beyond me. One might consider how perturbations and fluke switch points relate to specific types and biases of technical change. And one might state mathematical theorems and provide proofs.

More On Baldone Example

Figure 1: A Two-Dimensional Pattern Diagram, Enlarged

1.0 Introduction

This post further generalizes an example from Salvatore Baldone.. Like an example from Bertram Schefold, I find that Baldone's example is in a wedge near the edge of the appropriate region in one of my partitions of a parameter space. I have some very complicated spreadsheets that allow me to quickly visualize the effects of varying parameters. Baldone and Schefold were working long before Visicalc, Microsoft Excel, or LibreOffice. Finding these numeric examples must have been tedious.

I think Baldone created this example to illustrate recurrence of truncation. Recurrence of truncation does not necessarily require the recurrence of techniques, but in this example recurrence of truncation occurs with recurrence of techniques. I found it interesting that I could also find here a non-monotonic variation of the economic life of a machine, without recurrence of techniques or capital-reversing.

2.0 Technology

Tables 1 and 2 specify the processes available in this economy. In the first process, labor works with corn to produce a new machine. In the remaining three processes, labor works with corn and a machine to produce corn. A machine one year older than the machine used as an input is jointly produced with corn. Prices of production are defined for coefficients of production at a given moment in time. Technical progress leads to coefficients of production for inputs declining over time. The notation allows for technical progress to be at different rates in the machine and corn sectors.

Table 1: Inputs for The Technology

Input	Process
	(I)	(II)	(III)	(IV)
Labor	(2/5) e1 - σ t	(1/5) e1 - φ t	(3/5) e1 - φ t	(2/5) e1 - φ t
Corn	(1/10) e1 - σ t	(2/5) e1 - φ t	0.578 e1 - φ t	(3/5) e1 - φ t
New Machine	0	1	0	0
1-Yr. Old Machine	0	0	1	0
2-Yr. Old Machine	0	0	0	1

Table 2: Outputs for The Technology

Output	Process
	(I)	(II)	(III)	(IV)
Corn	0	1	1	1
New Machine	1	0	0	0
1-Yr. Old Machine	0	1	0	0
2-Yr. Old Machine	0	0	1	0

Three techniques of production exist here at each moment in time. I assume old machines can be discarded without cost. In the Alpha technique, machines are used for only one year. Old machines are used for two years in the Beta technique. In the Gamma technique, they are used for the full three years.

3.0 Prices of Production in Baldone Example

I start by reproducing Baldone's example. I assume labor is advanced, and workers are paid a wage at the end of the year. Corn is taken as the numeraire. For each technique, one can solve for the wage and prices of machines as a function of the rate of profits. Figure 2 plots the wage curves for each technique. The cost-minimizing technique, at a given rate of profits, maximizes the wage. That is, the wage is on the outer envelope. It is not very visually obvious which technique is cost-minimizing, so I have labeled the cost-minimizing techniques. And one sees the Alpha is technique is cost minimizing at a low rate of profits. Around the switch point between the Alpha and Beta techniques, a higher wage is associated with the adoption of a more labor-intensive techniques.

Figure 2: The Wage Frontier with the Recurrence of Techniques

An aspect of the choice of technique can be seen by looking at prices. Figure 3 shows the price of new machines. For all techniques, the price of a new machine is positive for any feasible distribution. At a switch point, the price of a new machine is the same for both techniques that are cost-minimizing. Figure 4 shows how the price of old machines varies with the rate of profits. If a technique is cost-minimizing, the prices of old machines produced by that technique are non-negative at that rate of profits. At a switch point, the price of at least one produced old machine is zero. Baldone's article goes into much detail about prices of machines and truncation.

Figure 3: The Price of a New Machine

The price of an old machine is zero for a technique in which that machine is not produced. Figure 4 shows the prices of only those old machines that are produced in the corresponding technique. In an analysis of the choice of technique with fixed capital, if the price of an old machine is negative at a given rate of profits, the cost minimizing technique must have the economic life of machine must be truncated. Consider, for example, rates of profits less than approximately 4 percent or greater than approximately 63 percent. The price of a one-year old machine under the Beta technique is negative. The prices of one-year old and two-year old machines under the Gamma technique are both negative. Thus, neither can be cost-minimizing. The Alpha technique must be cost-minimizing in these ranges of the rates of profits.

Figure 4: The Price of Old Machines in the Baldone Example

4.0 A Time Path

I now take a first step in generalizing Baldone's capitalism. The rate of decrease of coefficients of production happens to be ten percent in both the machine and corn sectors. Figure 5 shows how the wage frontier varies with time under these assumptions. The maximum rate of profits and the rate of profits at switch points are plotted against time.

Figure 5: Variation in the Wage Frontier with Time

Here, the economy is not viable at the initial time. There must be some sort of low-productivity, backstop technology that was previously used. Figure 5 partitions time into six regions, and Figure 6 enlarges transient regions. Baldone's example is in Region 9.

Figure 6: An Enlargement of the Variation in the Wage Frontier

5.0 A Partition of the Parameter Space

I now let the rates of decrease in coefficients of production differ between the machine and corn sector. Figure 7 graphs the resulting two-dimensional space and how it is partitioned by fluke switch points, which I call patterns. I only label one partition in Figure 7. For the partition between Region 0 and Region 1, the maximum rate of profits for the Alpha technique is zero, and the maximum rate of profits is negative for the Beta and Gamma techniques.

Figure 7: A Two-Dimensional Pattern Diagram

I look at two enlargements of parts ot the space in Figure 7 to get a somewhat more visually obvious understanding of what is going on here. Figure 8 is a blow-up of the middle left of Figure 7, and Figure 1 is a blow-up towards the middle right of Figure 7. I suppose I should say something more about this graph, but I will content myself with Table 3 and one observation. Consider the intersection of the boundaries between Regions 1 and 2, between 2 and 6, between 6 and 7, and between 7 and 1. This point is an intersection of three patterns over the wage axis with a three-technique pattern. I want to claim this intersection is generic, in some sense. I suppose precisely specifying in what sense would be publishable but maybe is beyond me.

Figure 8: An Enlargement of the Parameter Space

Table 3: Results

Region	Technique	Summary
0	None	Not viable.
1	Alpha	No switch points.
2	Beta, Alpha	The switch point exhibits negative real Wicksell effects. A smaller rate of profits is associated with a longer economic life of a machine.
3	Beta	No switch points.
4	Gamma, Beta	The switch point exhibits negative real Wicksell effects. A smaller rate of profits is associated with a longer economic life of a machine.
5	Gamma	No switch points.
6	Gamma, Beta, Alpha	Switch points exhibit negative real Wicksell effects. A smaller rate of profits is associated with a longer economic life of a machine.
7	Gamma, Alpha	The switch point exhibits negative real Wicksell effects. A smaller rate of profits is associated with a longer economic life of a machine.
8	Alpha, Gamma, Alpha	The switch point at the higher rate of profits exhibits positive real Wicksell effects. Reswitching of techniques and recurrence of truncation.
9	Alpha, Gamma, Beta, Alpha	The switch point between Beta and Alpha exhibits positive real Wicksell effects. Recurrence of techniques and of truncation.
10	Alpha, Gamma, Beta	Switch points exhibit negative real Wicksell effects. A smaller rate of profits is associated with a non-monotonic variation in the economic life of machine.
11	Alpha, Gamma	The switch point exhibits negative real Wicksell effects. A smaller rate of profits is associated with a shorter economic life of a machine.

6.0 Non-Monotonic Variation of the Economic Life of a Machine with the Rate of Profits

I might as well illustrate the wage frontier (Figure 9) in Region 10. From low to high wages (that is, high to low rates of profits) the cost-minimizing technique ranges from Beta through Gamma to Alpha. In a stationary state, the machine is run for two years at maximum rate of profits. At a middling rate of profits, its economic life is increased to three years. At an even lower rate, its economic life jumps down to one year. Around both switch points, corn produced per person-year is higher at the lower rate of profits. In some sense, the technique adopted at the lower rate of profits is more capital-intensive, despite the non-monotonic variation in the economic life of the machine. The switch point between Beta and Gamma is consistent with Austrian claims, but the switch point between Gamma and Alpha is a logical disproof of their capital theory.

Figure 9: The Wage Frontier in Region 10

For completeness, Figure 10 plots the prices of produced old machines by technique. For a rate of profits below approximately 10.7 percent, the price of a one-year old machine is negative for both the Beta and Gamma techniques. Thus, neither is cost-minimizing in this range; the Alpha technique is. Between approximately 10.7 and 110 percent, the prices of both one-year old and two-year old machines is positive under the Gamma technique. It is not cost-minimizing to truncate the machine to one or two years in this range. For even larger feasible rates of profits, the price of a two-year old machine is negative under the Gamma technique, and the price of a one-year old machine is positive under the Beta technique. In this range, it is cost-minimizing to operate the machine for two years.

Figure 10: Prices of Old Machines in Region 10

7.0 Conclusion

My methodology for generalizing Baldone's example leads to some complicated graphs. I find a couple of new phenomena that I have not seen in other examples of fixed capital. I think of Region 0, in which no specificed technique is viable. More interesting to me is Region 10, in which the economic life of a machine varies non-monotonically with the rate of profits, without either recurrence of techniques or cost-minimizing.

Reference

• Salvatore Baldone. 1974. Il capitale fisso nello schema teorico di Piero Sraffa. Studi Economici XXIV(1): 45-106. Translated in Pasinetti (1980).

An Extension Of An Example From Salvatore Baldone

Figure 1: A Pattern Diagram, Enlarged

1.0 Introduction

This post looks at and generalizes an example of the recurrence of techniques by Salvatore Barone. It is an example with fixed capital illustrating the recurrence of the period of truncation. In the generalization, I find what I call patterns over the axis for the rate of profits, a patern over the wage axis, a three-technique pattern, and a reswitching pattern.

Barone's example demonstrates that around a switch point, a lower rate of profits can be associated with both an increase and a decrease in the economic life of a machine and an increased life of a machine can be associated with both an increase and a decrease in the capital-intensity of a technique. From other examples, I know the variability in the direction of the period of truncation with the rate of profits, the (non) relationship of the economic life of a machine with output per worker, and the jump (from one years to three) in the economic life of a machine with an infinitesimal variation in the rate of profits are independent of reswitching and capital-reversing.

So much for the Austrian theory of capital.

2.0 Technology

The available technology consists of the four processes in Tables 1 and 2. Each process exhibits constant returns to scale (CRS) and takes a year to complete. In the first process, labor and corn are used to make a machine, which, I suppose, I could have called a tractor. In the remaining three processes, labor, corn, and the machine are used to make corn. In each of the first two of these three processes, a machine one year older than it was as an input is jointly produced with corn. Corn is circulating capital and the machine is fixed capital.

Table 1: Inputs for The Technology

Input	Process
	(I)	(II)	(III)	(IV)
Labor	a0, 1	a0, 2	a0, 3	a0, 4
Corn	a1,1	a1,2	a1,3	a1, 4
New Machines	0	1	0	0
1-Year Old Machines	0	0	1	0
2-Year Old Machines	0	0	0	1

Table 2: Outputs for The Technology

Output	Process
	(I)	(II)	(III)	(IV)
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 assume that an old machine can be costlessly disposed of before its technical life. Thus, there are three techniques that can arise in a stationary state. In the Alpha technique, the machine is junked after being used one year; only the first two processes are operated. In Beta, the machine is junked after two years. In Gamma, the machine is operated for its full technical life and all four processes are operated.

I conclude this section by specifying parameters for the coeffients of production:

a_{0, 1} = (2/5) e^{1 - t/10}

a_{0, 2} = (1/5) e^{1 - t/10}

a_{0, 3} = (3/5) e^{1 - t/10}

a_{0, 4} = (2/5) e^{1 - t/10}

a_{1, 1} = (1/10) e^{1 - t/10}

a_{1, 2} = (2/5) e^{1 - t/10}

a_{1, 3} = 0.578 e^{1 - t/10}

a_{1, 4} = (3/5) e^{1 - t/10}

Barone's example arises when t = 10. The exponential decay in these coefficients is a description of technical progress as exogeneous.

3.0 Prices of Production

I now want to consider prices of production, given the technology at a point of time, specifically for Baldone's example. I take corn as numeraire. For a given technique, each operated process provides an equation. I take labor as advanced and assume wages are paid out of the end of the year. Given the rate of profits, one can then solve for the wage and the prices of a new machine, a one-year old machine, and a two-year old machine.

Figure 2 plots the wage curves for the three techniques. (By the way, Baldone has a transcription error in at least one of his equations. I was able to replicate his tables with this error corrected.) The cost-minimizing techniques are noted, even though which is on the outer frontier is not always easily visible.

Figure 2: The Wage Frontier in Baldones Example

Figure 3 shows the price of a new machine. At a switch point point, prices are identical for the techniques whose wage curves intersect at that switch point. For example, a rate of profits of approximately 4 percent, the price of a new machine for the Alpha and the Gamma technique is the same.

Figure 3: The Price of a New Machine

Figure 4 is finally an example which is visually obvious. The price of an old machine is zero for a technique in which that machine is not produced. Figure 4 shows the prices of only those old machines that are produced in the corresponding technique. In an analysis of the choice of technique with fixed capital, if the price of an old machine is negative at a given rate of profits, the cost minimizing technique must have the economic life of machine must be truncated.

Figure 4: The Price of Old Machines

Consider, for example, rates of profits less than approximately 4 percent or greater than approximately 63 percent. The price of a one-year old machine under the Beta technique is negative. The prices of one-year old and two-year old machines under the Gamma technique are both negative. Thus, neither can be cost-minimizing. The Alpha technique must be cost-minimizing in these ranges of the rates of profits.

At a rate of profits of approximately 56 percent, the price of a one-year old is positive and the same for the Beta and Gamma techniques, and the price of a two-year old machine is zero under the Gamma technique. This is a switch point for the Beta and the Gamma technique. The price of a two-year old machine is negative under the Gamma technique for any rate of profits greater than this. The Gamma technique cannot be cost-minimizing between 56 and 63 percent. Since the price of a new machine and a one-year old machine is positive, in this range, under the Beta technique, it is not cost-minizing to truncate the machine to an economic life of one year. By the same logic, it is not cost-minimizing to truncate the machine at all for rates of profits between 4 percent and 56 percent.

The analysis of the choice of techniques based on prices yields the same conclusions as an analysis based on the construction as the outer wage frontier. Table 3 summarizes characteristics of the Baldone example. The switch point at approximately 56% is the only one of the three that is not 'perverse'. Around this switch point, a lower rate of profits is associated with an increase in the economic life of the machine, a greater capital-intensity, and more output per worker.

Table 3: Summary of Barone Example

0 ≤ r ≤ 4.066%	Alpha cost minimizing.
r ≈ 4.066%	Lower rate of profits associated with a decrease in the economic life of the machine, from three years to one year. Consumption per worker increased.
4.066% ≤ r ≤ 55.656%	Gamma cost-minimizing.
r ≈ 55.656%	Lower rate of profits associated with an increase in the economic life of the machine, from two to three years. Consumption per worker increased.
55.656% ≤ r ≤ 62.732%	Beta cost-minimizing.
r ≈ 62.732%	Lower rate of profits associated with an increase in the economic life of the machine, from one to two years. Consumption per worker decreased.
62.732% ≤ r ≤ 74.166%	Alpha cost minimizing.

4.0 An Extension for Structural Dynamics

I now introduce structural dynamics. I let all coefficients of production for inputs of labor and corn decrease exponentially, at a rate of 10%. Figure 5 illustrates how the variation of the cost-minimizing technique with the rate of profits changes with technical progress. Here, the economy is not viable at the initial time. There must be some sort of low-productivity, backstop technology that was previously used. Figure 5 partitions time into six regions, and Figure 1, at the top of this post enlarges transient regions. Baldone's example is in Region 3.

Figure 5: A Pattern Diagram

5.0 Conclusion

I suppose I should figure out a complete characterization of all six regions, not just Region 3. I now have more examples with fixed capital for my approach to structural economic dynamics than can be comfortably be described in a paper of reasonable length.

I am finding that the analysis of so-called 'paradoxes' in models of the prices of production with fixed capital is an important extension of the Cambridge Capital Controversy. A thorough understanding of the 13-page Chapter X in Sraffa's book only became available in English after mainstream economists had commenced on ignoring certain results.

References

• Salvatore Baldone. 1974. Il capitale fisso nello schema teorico di Piero Sraffa. Studi Economici XXIV(1): 45-106. Translated in Pasinetti (1980).

Political Novels?

I would like suggestions to add to this list:

• Benjamin Disraeli, Coningsby or the New Generation.
• Anthony Trollope, The Way We Live Now.
• Allen Drury, Advise and Consent: A Novel of Washington Politics.
• John Ehrlichman, The Company.
• Anonymous (Joel Klein) Primary Colors: A Novel of Politics.

This is not for Christmas, but some of my personal reading. I am aware that Coningsby is the first of a trilogy, that Advise and Consent is the first of a series, and that Primary Colors has a sequel. Ehrlichman's novel did not make a lasting impression on me. As usual, I find it hard to define what I think groups these together.

Disraeli writing his novels in the midst of trying to climb the greasy pole is hard to fathom:

"The Duke talks to me of Conservative principles; but he does not inform me what they are. I observe indeed a party in the State whose rule it is to consent to no change, until it is clamorously called for, and then instantly to yield; but those are Concessionary, not Conservative principles. This party treats institutions as we do our pheasants, they preserve only to destroy them. But is there a statesman among these Conservatives who offers us a dogma for a guide, or defines any great political truth which we should aspire to establish? It seems to me a barren thing, this Conservatism, an unhappy cross-breed; the mule of politics that engenders nothing." -- Disraeli

Triple Switching and Fluke Switch Points

Figure 1: A Pattern Diagram with Triple Switching

In demonstrating the lack of foundation for claims of the Austrian school about the supposed relationships between a greater supply of capital, a consequent lower rate of profits, and a longer period of production, I have so far only presented examples in which the economic life of an existing machine can be extended or truncated. Schefold (1980: 170) interprets a more roundabout technique as one in which a long-lived machine is used to produce a finished good that previously was produced directly without the aid of fixed capital or, at least, with a different and inferior machine. The example in this post extends Schefold's illustration of the difficulty in sustaining the Austrian claim.

I am disappointed that in briefly exploring the parameter space specified by coefficients of production, I was unable to find an example in which wages curves on the frontier were easily distinguishable by the eye. I did like that Figure 1 came out one way I knew could bring about triple switching.

The second, third, and fourth processes in the technology (Tables 1 and 2) constitute the corn sector. In Process II, corn is produced from inputs of labor and corn, without fixed capital. The Alpha technique (Table 3) consists of Process II alone. A machine sector, composed of Process I, exists in the Beta and Gamma techniques. The technical life of the machine is two years. The machine is truncated to one year in the Beta technique

Table 1: Inputs for The Technology

Input	Process
	(I)	(II)	(III)	(IV)
Labor	a0,1	a0,2	a0,3	a0,4
Corn	a1,1	a1,2	a1,3	a1,4
New Machines	0	0	1	0
Old Machines	0	0	0	1

Table 2: Outputs for The Technology

Output	Process
	(I)	(II)	(III)	(IV)
Corn	0	1	b1,3 = 1/2	b1,4 = 1/2
New Machines	1	0	0	0
Old Machines	0	0	1	0

Table 3: Techniques

Techniques	Processes
Alpha	(II)
Beta	(I), (III)
Gamma	(I), (III), (IV)

Suppose the coefficients of production for inputs of labor and circulating capital decrease ten percent per year. Figure 1 shows the variation in the choice of technique for a specific configuration of coefficients of production. Schefold's example of triple switching occurs at t = 10. Figure 2 graphs the wage frontier here, in which the wage curves for the Alpha and Gamma techniques are difficult for the eye to distinguish. Figure 3 shows extra profits in operating the second process at Gamma prices. Triple switching is more apparent here.

Figure 2: Wage Frontier for Triple Switching

Figure 3: Extra Profits at Gamma Prices

With this specification of parameters and technical progress, the non-roundabout process Alpha eventually replaces the roundabout process Gamma, whatever the distribution of income. Triple switching appears in the middle of the three transient regions. The patterns of switch points illustrate one manner in which triple-switching can appear. If the pattern over the wage axis were to arise before the second reswitching pattern, the region in which triple-switching occurs would be followed by a region with double switching. With more techniques, one of the switch points could be replaced on the frontier in a three-technique pattern, instead of eventually vanishing over an axis. At any rate, this example continues to illustrate how combinations of patterns of switch points can illuminate the effects of technical change.

In Regions 2, 3, and 4, the Alpha technique, in which corn is produced without the use of a machine, is cost-minimizing at the highest rate of profits, where the wage is zero. Around the only switch point in each of Regions 2 and 4, a lower rate of profits is indeed associated with a more roundabout technique, and the more roundabout technique has a higher level of consumption per person-year in a stationary state. The Austrian claim is also illustrated at the lowest and highest switch point in Region 2. But it is invalidated for the middle switch point. Around this switch point, a lower rate of profits is associated with the replacement of a roundabout technique by the direct production of the consumer commodity.

This numerical example re-iterates that no necessary connection exists between employing or lengthening the economic life of a machine and an increase in the use of 'capital'. Bohm-Bawerk (1959) was incorrect not merely because of the difficulty of defining a quantitative measure of the average period of production. His intuition, and not just his, on how prices work was itself incorrect.