**Summary:**

Exploring perturbations of four examples of fluke switch points provides a brief survey of some aspects of prices of production. The examples arise in, respectively, models of circulating capital, fixed capital, extensive rent, and intensive rent. The reverse substitution of labor, reswitching, and capital reversing, for example, are contrasted with genuine fluke cases. These posts present examples of fluke switch points. Each example is of a fluke case in at least two ways. Either two switch points are flukes, or a switch point exhibits two fluke properties. For example, one fluke switch point might be on the axis for the rate of profits and another might be on the wage axis. I say a switch point is a fluke if it is a knife edge case in which almost all perturbations of model

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Robert Vienneau considers the following as important: Example in Mathematical Economics, Joint Production, Sraffa Effects

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Exploring perturbations of four examples of fluke switch points provides a brief survey of some aspects of prices of production. The examples arise in, respectively, models of circulating capital, fixed capital, extensive rent, and intensive rent. The reverse substitution of labor, reswitching, and capital reversing, for example, are contrasted with genuine fluke cases.

These posts present examples of fluke switch points. Each example is of a fluke case in at least two ways. Either two switch points are flukes, or a switch point exhibits two fluke properties. For example, one fluke switch point might be on the axis for the rate of profits and another might be on the wage axis. I say 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.

This analysis builds on post-Sraffian price theory. Kurz & Salvadori (1995) provide a comprehensive textbook treatment of the analysis of prices of production and of the choice of technique, including in models with circulating capital, fixed capital, and extensive and intensive rent. Parameter spaces for fluke switch points are explored by Vienneau (2021 and 2022). Prices of production, in competitive markets, are such that operated processes can satisfy requirements for use after replacing commodity inputs required for production. A single rate of profits is made in all operated processes, and no extra profits are obtainable in non-operated processes.

National income and product accounts (NIPAs) provide Leontief matrices with elements expressed in price ratios. Given price indices by industry or normalization methods, one can obtain Leontief matrices in physical terms. These matrices, with certain abstractions, can be used to find prices of production corresponding to any functional distribution of income. Price flows include vintage technologies with old plants earning quasi rents. Fixed capital is hard to handle rigorously in empirical data (Kurz 2021). The rate of profits cannot be expected to be uniform across industries, since competitive conditions do not always prevail. Be that as it may, a problem of the choice of technique arises by combining Leontief matrices across time or countries. Han & Schefold (2006) and Zambelli (2018), following roughly the above approach, find some examples, albeit not many, of reverse labor substitution, of the reswitching of techniques, and of capital reversing.

Given the knife edge property of the fluke cases presented in the article, they cannot be expected to be observed in the empirical data. But local perturbations of fluke cases point to the possibility of qualitative differences in the analysis of the choice of technique. Exploring parameter spaces near selected fluke cases facilitates a brief survey of some aspects of prices of production.

These posts do not explore how such qualitative change impacts the temporal dynamics of market prices, an unsolved problem in price theory (Kurz & Salvadori 2022, Bellino 2011). If one accepts that prices of production, given technology, net output, and a distributive variable, provide information on tendencies in market prices, perturbations of coefficients of production indicate qualitative changes in such tendencies. In examples or reverse labor substitution and of capital reversing, higher wages are associated with the adoption of a technique in which more labor is hired, respectively, per unit gross output in a given industry and per unit net output in the economy overall. But variations in coefficients of production may eliminate such possibilities. Perturbing fluke cases in numerical examples suggests which specific qualitative changes are possible.

These posts also do not draw conclusions about the probability of, for example, capital-reversing arising in actual data. Fluke cases correspond to lower-dimensional manifolds in multi-dimensional parameter spaces. Regions in the parameter space corresponding to reswitching, for example, are of strictly positive measure. I am reluctant to draw any conclusions from the relative sizes, in these examples, of regions formed by partitions from fluke cases. Schefold (2022) notes that given random matrices of coefficients of production, reswitching is rare. Very few techniques, however, lie on the wage-rate of profits frontier. His argument critiquing marginalist economics along these lines seems independent of the approach in these posts.

This post is about four other posts. The first explores an example of two fluke switch points in a model of circulating capital. The reverse substitution of labor results from a perturbation of this fluke case. The second post presents an example of a fluke switch point in a model of fixed capital. Reswitching emerges with perturbations of selected parameters The third post describes two fluke switch points in a model of extensive rent. The order of fertility differs from the order of rentability with appropriate perturbations. The fourth post presents a fluke switch point in a model of intensive rent and discusses the emergence of the non-existence and non-uniqueness of a cost-minimizing technique.

Many more fluke cases, some of different types, can be constructed for those with the requisite patience. The reverse substitution of labor, reswitching, capital reversing, the association of the truncation of the economic life of a machine with the choice of a more capital-intensive technique, a divergence between the order of fertility and the order of rentability, the variation in the existence of rent with the rate of profits, and the non-uniqueness and the non-existence of a cost-minimizing technique are not fluke cases. This article demonstrates this result by contrasting these possibilities with genuine fluke cases.