An Example Of Fixed Capital From Salvatore Baldone

Figure 1: Wage Curves For A Technique In The Example

1.0 Introduction

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 Flows

Each 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.

Table 1: Inputs for The Technology

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

Table 2: Outputs for The Technology

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 Prices

In 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:

• For Alpha, when 0 < r < 74.2 percent
• For Beta, when 0 < r < 73.8 percent
• For Gamma, when 0 < r < 72.7 percent.

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.

Figure 2: Prices of New Machines

One-year old machines have positive prices (Figure 3):

• For Beta, when 43.6 percent < r < 62.7 percent
• For Gamma, when 4.1 percent < r < 56.9 percent

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,

Figure 3: Prices of One-Year Old Machines

Two-year old machines have positive prices (Figure 4):

• For Gamma, when 0 < r < 55.7 percent

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.

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:

• Alpha, for 0 < r < 4.1 percent and 62.7 percent < r < 74.2 percent
• Beta, for 55.7 percent < r < 62.7 percent
• Gamma, for 4.1 percent < r < 55.7 percent

Now, I can look at what happens around the three switch points:

• Around r = 62.7 percent, a lower interest rate is associated with a switch from Alpha to Beta, a more roundabout technique. But net output per worker falls. A more roundabout technique is less capital-intensive.
• Around r = 55.7 percent, a lower interest rate is associated with a switch from Beta to Gamma, a more roundabout technique. And net output per worker rises.
• Around r = 4.1 percent, a lower interest rate is associated with a switch from Gamma to Alpha, a less roundabout technique. And net output per worker rises. A less roundabout technique is more capital-intensive.

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 Conclusion

The 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.

References

• Baldone, Salvatore (1974), Il capitale fisso nello schema teorico di Piero Sraffa, Studi Economici, XXIV(1): 45-106. Trans. in Pasinetti (1980).
• Kurz, Heinz D. and Neri Salvadori. 1995. Theory of Production: A Long-Period Analysis. Cambridge: Cambridge University Press.
• Pasinetti, Luigi L., (1980) (ed.), Essays on the Theory of Joint Production, New York: Columbia University Press

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

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

When Did The Marginalist Theory Of Labor Markets Become Obsolete?

Chemists once believed, before Lavoisier and Priestly discovered oxygen, in the theory of phlogiston. Physicists, before Galileo, believed in the impetus theory of motion. Academic economists once believed that, in competitive markets, wages and employment tend to the point of intersection of supply and demand curves. The supply curve is supposed to slope up, showing that with a higher real wage, the hours offered for employment increase. The demand curve slopes down, modeling a smaller quantity demanded of labor services at higher wages. A short-run and long-run version of the theory existed.

When did this theory become obsolete? Some candidates:

• With the 2015 publication of Arrigo Opocher and Ian Steedman's Full Industry Equilibrium.
• With Pierangelo Garegnani's 1970 paper, Heterogeneous capital, the production function and the theory of distribution.
• With Piero Sraffa's 1960 book, The Production of Commodities by Means of Commodities.
• With the 1936 publication of John Maynard Keynes' The General Theory of Employment, Interest qand Money.
• With G. F. Shove's 1933 review of J. R. Hicks' The Theory of Wages.

Empirical work went along with this timeline, whether that includes the discovery that firms use markup pricing or the use of natural experiments showing that minimum wages do not decrease employment.

Oh, what’s that you say? You have not heard that the theory of supply and demand is obsolete? Well, not everybody can be expected to understand the periodic table or laws of motion.

Technological Progress In The Production Of Type II Tractors

Figure 1: Variation in the Choice of Technique with Technical Progress

This post is a continuation of an analysis of an example in the corn-tractor model.

An analysis of technical change is another application of this partitioning of parameter space by fluke switch points. In this context, the change in properties of the wage frontier is the result of structural economic dynamics (Pasinetti 1993). A movement from the upper right to the lower left in Figure 2 reflects a specific kind of technical progress in producing tractors of type II. The quantity of type II tractors needed to manufacture a new type II tractor decreases with a movement to the left. The maximum rate of profits for the technique with type II tractors increases. The quantity of type II tractors needed to make a bushel corn decreases with a downward movement. The maximum wage increases.

Figure 1 depicts the variation in the analysis of the choice of technique along a specific line in Figure 2. The coefficients of production for inputs of type II tractors, in both industries, fall together in travelling from the upper-right to the lower-left in the graph. Switch points and the maximum wage are plotted. Four fluke switch points provide vertical divisions in the diagram. The cost-minimizing technique is labeled among ranges of the wage. From right to left, the diagram shows how technical progress in producing type II tractors results in the corresponding technique ultimately replacing the technique with type I tractors.

Region 6, with triple-switching, arises in the midst of this transition. It can be preceded by a region with a single switch point, as it is in this diagram. Or triple-switching can be preceded by region 3, in which double-switching occurs. The region with triple-switching can be followed by region 7, with a single switch point, or by region 5, with double-switching. A region with triple-switching might not occur at all, as in a path through regions 3, 2, and 5. The appendix provides an example in which double-switching can occur, but not triple-switching. Double-switching might also not occur at all, with a path from region 4, through region 2, and into region 1.

This example therefore suggests that a technology in which the corresponding prices of production exhibit triple-switching will appear only as a transient phenomenon, as one technique replaces another due to technical change. This conclusion also applies to double-switching. Presumably, the same result applies to multiple switching with four or more switch points.

How multiple switching manifests in market prices depends on many determinations not considered in this article. The speed at which coefficients of production evolve with technical change, compared to the speed at which market prices approach prices of production, if they do, seems of some importance. Robinson (1975) argues the latter process should be analyzed in historical time, not with a mechanical process in logical time. The stability of wages and the rate of profits is another issue. Accounting conventions for depreciation and for allocating overhead costs might impact these processes. The size of extra profits obtained by being first to adopt a new process or technique is another consideration. Nevertheless, no theoretical basis seems to exist for the idea, for example, that a rise in wages or a fall in the interest rate is associated with a drop in employment due to the adoption of a less labor-intensive or more capital-intensive technique of production, out of a given and known book of blueprints.

Publicly Available Matlab Code For The Analysis Of The Choice Of Technique

I have created a project on GitHub: SraffianAnalysis.

This directory contains Octave code to support the analysis of the choice of technique.

I have examples of code to specify discrete technologies for small economies and to plot wage curves and switch points for them. Techniques in these economies produce one, two, three, or four commodities. Switch points, which are found as the roots of certain polynomial equations, can be found exactly. I allow the techniques to be specified as general joint production and with persistent differentials in the rate of profits. (The cost-minimizing technique in general joint production is not necessarily found as on the outer envelope of wage curves.)

I also have some main programs to find fluke switch points, like a switch point that is an intersection of more than two wage curves. A Microsoft Excel spreadsheet and PowerPoint slide decks illustrate what can be done with this capability.

Recent Work On The Economic Calculation Problem

This post is mostly a bibliography. I know that Ludwig Von Mises' argument that socialist central planning will not work is invalid.

Many have put forth plans for post-capitalist societies. W. Paul Cockshott and Allin Cottrell's Towards a New Socialism is interesting in that they consider issues of computational complexity. I also once read some work of their colleagues Greg Michaelson and Ian Wright.

Apparently, a flurry of recent research investigates whether or not improvements in computing technology refute the Austrian argument against central planning. I have read hardly anything in the following list:

• Boettke, Peter J., and Rosolino A. Candela. 2023. On the feasibility of technosocialism. Journal of Economic Behavior and Organization 205: 44–54.
• Peter Boettke, Rosolino A. Candela, and Tegan L. Truitt. 2024. The Socialist Calculation Debate.
• Paul F. Cwik and Lucas M. Engelhardt. 2023. Revisiting the computation problem. Quarterly Journal of Austrian Economics. 26(3): 325-347.
• Dapprich, Jan Philipp. 2023. Optimal planning with consumer feedback: a simulation of a socialist economy. Review of Political Economy 35 (4): 1136–56.
• Dapprich, Jan, and William Paul Cockshott. 2023. Input-output planning and information. Journal of Economic Behavior and Organization 205:412–22.
• Dapprich, Jan, and Dan Greenwood. 2024. Cybersocialism and the future of the socialist calculation debate. Erasmus Journal of Philosophy and Economics 17 (1): 1–23.
• Lambert, Karras J, and Tate Fegley. 2023. Economic calculation in light of advances in big data and artificial intelligence. Journal of Economic Behavior and Organization 206 (February): 243–50.
• Nieto, Maxi. 2022. Entrepreneurship and decentralised investment in a planned economy: a critique of the Austrian reading. Historical Materialism 30, no. 1: 1–31.
• Nieto, Maxi, and Juan Pablo Mateo. 2020. Dynamic efficiency in a planned economy: innovation and entrepreneurship without markets. Science and Society 84, no. 1 (January): 42–66.
• Nguyen, Hai-Trieu v. 2024. The incompleteness of central planning. Quarterly Journal of Austrian Economics. 27(4): 1-22
• Phelan, Steven E, and Nikolai G Wenzel. 2023. Big data, quantum computing, and the economic calculation debate: will roasted cyberpigeons fly into the Mouths of comrades? Journal of Economic Behavior and Organization 206: 172–81.

Austrian Capital Theory And Triple-Switching In The Corn-Tractor Model

Table 1: Lower Rate of Profits around a Switch Point

	Tradional Marginalist Story	'Perverse' Marginalist Story
Traditional Austrian Story	Greater net output per worker	Smaller net output per worker
	More roundabout technique	More roundabout technique
	Switch pt. in region 2, 1st in region 5	2nd switch point in region 3, 2nd in region 6
'Perverse' Austrian Story	Greater net output per worker	Smaller net output per worker
	Less roundabout technique	Less roundabout technique
	1st in region 3, 1st and 3rd in region 6, switch point in region 7	2nd switch point in region 5

My examination of triple-switching in the corn-tractor model allows for drawing some conclusions about Austrian capital theory.

The corn-tractor model, like the Samuelson-Garegnani model, is useful for investigating certain aspects of capital-theory. In obsolete theory from economists of the Austrian school, capital-intensity is associated with roundaboutness (Hennings 1987). A more roundabout technique is identified here with the use of a tractor with a longer lifetime. Only cases in which each type of tractor lasts for the same time in both industries are considered in this article. Thus, the degree of roundaboutness is unambiguous here. In the Austrian theory, a more roundabout technique, in a comparison of stationary states, is supposed to result in a greater net output per worker.

In a stationary state, tractors of each age are operated in parallel, both in the tractor industry and in the corn industry. At the end of each year, the oldest tractors are discarded and the appropriate number of new tractors are added to the stock. The sum of the prices of production of the stock of tractors is the value of capital. Following Steedman, I take a non-physical measure of capital-intensity to be the ratio of the value of capital to the value of net output. The capital-output ratio is a dimensionless number, while the units for the ratio of the value of capital to employment depends on the choice of the numeraire. In a stationary state, net output consists solely of corn, which is consumed. Net output per worker is an unambiguous physical quantity here.

For completeness, I repeat my summary (Table 2) of the analysis of the choice of technique in various regions.

Table 2: Cost-Minimizing Techniques by Region

Region	Cost-Minimizing Technique	Notes
1	Type II	No switch point. Type II tractors are dominant with sufficiently low coefficients of production in producing Type II tractors.
2	Type II, Type I	Around the switch point, a lower rate of profits is associated with a more roundabout technique, a greater capital-output ratio, and more consumption per person-year.
3	Type I, Type II, Type I	Around the first switch point, a lower rate of profits is associated with a less roundabout technique, a higher capital-output ratio, and more consumption per person-year. Around the second switch point, a lower rate of profits is associated with a more roundabout technique, a lower capital-output ratio, and less consumption per person-year.
4	Type I	No switch point. Type I tractors are dominant with sufficiently high coefficients of production in producing Type II tractors.
5	Type II, Type I, Type II	Around the first switch point, a lower rate of profits is associated with a more roundabout technique, a greater capital-output ratio, and more consumption per person-year. Around the second switch point, a lower rate of profits is associated with a less roundabout technique, a lower capital-output ratio, and less consumption per person-year.
6	Type I, Type II, Type I, Type II	Around the first and third switch point, a lower rate of profits is associated with a less roundabout technique, a greater capital-output ratio, and more consumption per person-year. Around the second switch point, a lower rate of profits is associated with a more roundabout technique, a smaller capital-output ratio, and less consumption per person-year.
7	Type I, Type II	Around the switch point, a lower rate of profits is associated with a less roundabout technique, a greater capital-output ratio, and less consumption per person-year.

Marginalist economists typically thought of prices as scarcity indices. A higher price of an input into production supposedly signals to mangers of firms to adopt processes in which now cheaper resources are substituted for that input.

"Assume that somewhere ... a new opportunity for the use of some raw material, say, tin, has arisen, or that one of the sources of supply of tin has been eliminated. It does not matter for our purpose ... which of these two causes has made, tin more scarce. ... If only some of [the users of tin] know directly of the new demand, and switch resources over to it, and if the people who are aware of the new gap thus created in turn fill it from still other sources, the effect will rapidly spread throughout the whole economic system and influence not only all the uses of tin but also those of its substitutes and the substitutes of these substitutes, ... without the great majority of those instrumental in bringing about these substitutions knowing anything at all about the original cause of these changes... The mere fact that there is one price for any commodity ... brings about the solution which (it is just conceptually possible) might have been arrived at by one single mind possessing all the information which is in fact dispersed among all the people involved in the process.” (Hayek 1948: 85-86)

This concept of the role of prices is undermined by the Cambridge capital controversy

Bliss (1975), in arguing for general equilibrium theory as an apposite response rejects this role of prices, at least when comparing equilibria:

"Even people who have made no study of economic theory are familiar with the idea that when something is more plentiful its price will be lower, and introductory courses on economic theory reinforce this common presumption with various examples. However, there is no support from the theory of general equilibrium for the proposition that an input to production will be cheaper in an economy where more of it is available. All that the theory declares is that the price of the use of an input which is more plentiful cannot be higher if all other inputs, all other outputs and all other input prices are in constant proportions to each other."

Suppose the rate of profits were an index for the scarcity of capital. A lower rate of profits would indicate that capital was more plentiful, in some sense, as compared to labor. Following the ideas of economists of the Austrian school, managers of firms would be encouraged to adopt more roundabout processes at a lower rate of profits around a switch point (Table 1). According to traditional marginalist reasoning, they would adopt a technique, at a lower rate of profits, with a higher capital-output ratio and more consumption per worker. The switch point in region 2 and the first switch point in region 5 are the only switch points that conform to these outdated ideas.

Other switch points illustrate that these ideas cannot be sustained in general. Consider the first switch point in region 3, the first and third switch points in region 6, and the switch point in region 7. Around these switch points, a lower rate of profits is associated with a higher capital-output ratio and more consumption per worker. Traditional marginalist reasoning is still validated. But, contrary to the expectations of economists of the Austrian school, a less roundabout technique is adopted. As shown in region 7, the disconnection between roundaboutness and capital-intensity does not even require reswitching for its demonstration.

The second switch points in regions 3 and 6, on the other hand, conform to Austrian but not to marginalist reasoning. Around these switch points a lower rate of profits is associated with a more roundabout technique, a lower capital-output ratio, and less consumption per person-year. More roundabout techniques need not be associated with greater capital-intensity or greater net output per worker.

Yeager (1979), in trying to justify Austrian theory with a concept of waiting, expresses puzzlement:

"One paradox not cleared up to my full satisfaction concerns consumption… Since this consumption paradox is a direct arithmetical implication of paradoxes already cleared up, and in particular of capital reversal or perversity, one might contend that no paradox remains. Yet this remark is not wholly satisfying."

The second switch point in region 5 contradicts both Austrian and marginalist reasoning. Around this switch point, a lower rate of profits is associated with a less roundabout technique. And it is associated with a lower capital-output ratio and less consumption per person-year. Perturbing parameters for a single example of triple-switching illustrates a variety of the so-called paradoxes discovered during the Cambridge capital controversy.

Robert Lucas On Recessions As Workers Choosing To Take Long Vacations

Why, under capitalism, do periods of persistent unemployment arise? Robert Lucas says the problem is to explain why workers do not to want to work:

"A theory that does deal successfully with unemployment needs to address two quite distinct problems. One is the fact that job separations tend to take the form of unilateral decisions - a worker quits, or is laid off or fired - in which negotiations over wage rates play no explicit role. The second is that workers who lose jobs, for whatever reason, typically pass through a period of unemployment instead of taking temporary work on the 'spot' labor market jobs that are readily available in any economy. Of these, the second seems to me the more important: it does not 'explain' why someone is unemployed to explain why he does not have a job with company X. After all, most employed people do not have jobs with company X either. To explain why people allocate time to a particular activity - like unemployment - we need to know why they prefer it to all other available activities: to say that I am allergic to strawberries does not 'explain' why I drink coffee. Neither of these puzzles is easy to understand within a Walrasian framework, and it would be good to understand both of them better, but I suggest we begin by focusing on the second of the two." -- Robert E. Lucas, Jr. 1987.Models of Business Cycles. Basil Blackwell: 53-54.

I suppose Lucas is to be commended that a regular, recurring relationship between employer and emplyee does not exist in the Walrasian model. Workers are auctioning off a supply of labor services at specific points in time, and no reason exists in the Arrow-Debreu model why those buying a specific agent's labor services today will have any tendency to hire the same agent's labor services tomorrow. But that bit about workers choosing to remain unemployed?

Other economists offer explanations as imperfections and frictions interfering in the operation of 'free' markets. George Akerlof explains unemployment by a social custom that wages must be 'fair'. Oliver Hart and others explain unemployment through employers having a better understanding of the worker's marginal product than the worker does. Others point to principal agent problems and information asymmetries.

John Maynard Keynes had a different approach. He explicity rejected explaining unempoyment by frictions:

"the classical theory has been accustomed to rest the supposedly self-adjusting character of the economic system on an assumed fluidity of money-wages; and, when there is rigidity, to lay on this rigidity the blame of maladjustment...

...The generally accepted explanation is, as I understand it, quite a simple one. It does not depend on roundabout repercussions, such as we shall discuss below. The argument simply is that a reduction in money-wages will cet. par. stimulate demand by diminishing the price of the finished product, and will therefore increase output and employment up to the point where the reduction which labour has agreed to accept in its money-wages is just offset by the diminishing marginal efficiency of labour as output (from a given equipment) is increased...

It is from this type of analysis that I fundamentally differ; or rather from the analysis which seems to lie behind such observations as the above. For whilst the above fairly represents, I think, the way in which many economists talk and write, the underlying analysis has seldom been written down in detail." -- John Maynard Keynes. 1936. The General Theory of Employment, Interest, and Money

To make sense of Keynes, a need arises for a price theory that is consistent with non-clearing labor markets. As some have been saying for decades, prices of production provide such a theory.

Perturbations Of Selected Parameters In The Corn-Tractor Model

Figure 1: Partitioning a Part of the Parameter Space with Fluke Cases

1.0 Introduction

This post is a continuation of the first example here. I examined a perturbation of two parameters of that example. I ended up with a more perspicacious partition of the parameter space than here.

2.0 Technology

Table 1 merely repeats the parameters for the fluke case that I started with. This case has switch points on the axis for the rate of profits and on the wage axis. A third switch point exists at an intermediate rate of profits.

Table 1: Parameters for Technology for First Example

Parameter	Type I Tractors	Type II Tractors
Tractor input per tractor (a)	≈ 0.306226	2/5
Labor input per tractor (b)	≈ 233.6967	20
Years tractors last in tractor industry (n)	1	2
Tractor input per bushel corn (α)	1	20
Labor input per bushel corn (β)	αI bI/aI	850
Years tractors last in corn industry (ν)	1	2

3.0 Perturbations of Selected Parameters

Almost any perturbation of the model parameters destroys fluke properties of the example in the previous section. Figure 1 illustrates perturbations in a_II and α_II. A switch point is on the axis for the rate of profits only for a specific value of a_II. Likewise, a switch point is on the wage axis only for the depicted partition of the parameter space, of, for instance, regions 1 and 7. The example in the previous post has parameters found at the intersection of these two partitions. The two other partitions occur for parameter values at which a switch point is repeated and the two wage curves are tangent at this switch point. The regions bounded by these partitions of the selected part of the parameter space are numbered.

The dashed line depicts the combination of coefficients of production for which the ratio of labor to tractors does not vary between industries for tractors of type II. To the left and above this line the physical capital-intensity of production is less, for type II tractors, in producing new tractors than it is in producing corn. To the right and below, the tractor industry for type II tractors has a larger physical capital-intensity than corn production.

I would have liked to have drawn the partitions as three-dimensional manifolds in a four-dimensional space, where (a_II, b_II, α_II, β_II) is a point in the space. But I can visualize a tesseract only momentarily, if at all (Heinlein 1941). Figure 1 is constructed by selecting only two parameters to perturb.

Double-fluke cases arise at intersections of the partitions. The partition between regions 2 and 3 is tangent to the partition between regions 3 and 4 at their point of intersection. Similarly, the partition between regions 1 and 5 is tangent to the partition between regions 2 and 5. The two partitions between regions 6 and 7 are tangent at their point of intersection, as well.

This last double-fluke switch point can perhaps admit of some elaboration. Figure 2 shows the rate of profits and the wage at switch points for each of two fluke cases. The solid lines correspond to the partition between regions 1 and 5 and the lower partition in Figure 2 between regions 6 and 7. The dashed lines correspond to the partition between regions 3 and 4 and the upper partition. Three switch points exist for the parameters along these two partitions. Two of these switch points are repeated roots, which is the fluke case under consideration. All three switch points coincide on the wage frontier at the double-fluke switch point to the extreme left.

Figure 2: The Rate of Profits at Switch Points with Tangent Wage Curves

Table 2: Cost-Minimizing Techniques by Region

Region	Cost-Minimizing Technique	Notes
1	Type II	No switch point. Type II tractors are dominant with sufficiently low coefficients of production in producing Type II tractors.
2	Type II, Type I	Around the switch point, a lower rate of profits is associated with a more roundabout technique, a greater capital-output ratio, and more consumption per person-year.
3	Type I, Type II, Type I	Around the first switch point, a lower rate of profits is associated with a less roundabout technique, a higher capital-output ratio, and more consumption per person-year. Around the second switch point, a lower rate of profits is associated with a more roundabout technique, a lower capital-output ratio, and less consumption per person-year.
4	Type I	No switch point. Type I tractors are dominant with sufficiently high coefficients of production in producing Type II tractors.
5	Type II, Type I, Type II	Around the first switch point, a lower rate of profits is associated with a more roundabout technique, a greater capital-output ratio, and more consumption per person-year. Around the second switch point, a lower rate of profits is associated with a less roundabout technique, a lower capital-output ratio, and less consumption per person-year.
6	Type I, Type II, Type I, Type II	Around the first and third switch point, a lower rate of profits is associated with a less roundabout technique, a greater capital-output ratio, and more consumption per person-year. Around the second switch point, a lower rate of profits is associated with a more roundabout technique, a smaller capital-output ratio, and less consumption per person-year.
7	Type I, Type II	Around the switch point, a lower rate of profits is associated with a less roundabout technique, a greater capital-output ratio, and less consumption per person-year.

The analysis of the choice of technique is qualitatively invariant in each numbered region. Table 2 lists the cost-minimizing technique, in order of an increasing rate of profits, in each region. One technique is cost-minimizing, whatever the distribution of income, in regions 1 and 4. One switch point exists in regions 2 and 7. Double-switching occurs in regions 3 and 5. Finally, triple-switching occurs in region 6. Perturbations of the parameters for an example in the previous post can result in each type of tractor being cost-minimizing in two discrete ranges of the wage or the rate of profits. This partitioning is not unique to this model.

4.0 Conclusions

I can examine specific properties of the switch points in each region, and maybe draw some more conclusions. But that will be for future posts.