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.
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