This blog, over years, presents a welter of fluke cases. I created many of the numerical examples to illustrate the reswitching of techniques, capital reversing, or some such so-called 'perversity'. Fluke cases can be combined. For example, a fluke switch point at a rate of profits of zero can also be a fluke switch point at which three wage curves intersect. Or two switch points on the wage frontier can both be fluke switch points at which four wage curves, not necessarily the same, intersect. Numerical examples remain to be developed for some possibilities.

Pattern of switch points over the wage axis |

Pattern of switch points for the reverse substitution of labor |

Pattern of switch points over the axis for the rate of profits |

Reswitching pattern of switch points |

Three-techniques pattern of switch points |

Four-technique pattern of switch points |

Pattern of switch points for the w-order of fertility |

Pattern of switch points for the r-order of fertility |

Pattern over the wage axis for the order of rentability |

Pattern over the axis for the rate of profits for the order of rentability |

Pattern for the requirements for use |

The analysis and construction of fluke cases yields insights into the analysis of the choice of technique in the system of prices of production. The reswitching of techniques, capital-reversing, process recurrence, the reverse substitution of labor, the extension of the lifetime of a machine at a lower wage, and the divergence between the order of fertility and the order of rentability are not fluke cases. These possibilities can be contrasted with genuine fluke cases, in which the perturbation of parameters destroys characteristics specifying such cases. These fluke cases partition parameter spaces into regions where these so-called 'perverse' phenomena arise.

These phenomena are also more or less independent of one another. Reswitching does not occur without process recurrence, but process recurrence can arise without reswitching. The association of a smaller rate of profits around a switch point with the truncation of the lifetime of a machine may or may not be accompanied by capital-reversing. Capital reversing can arise with or without a reverse substitution of labor and vice versa. Variations in the order of fertility need not accompany variations in the order of rentability. Nor need variations in the order of rentability be accompanied by variations in the order of fertility. The divergence between the order of fertility and the order of rentability can arise in an example of the reswitching of techniques, but reswitching is not necessary for such divergences. These specific examples do not exhaust the possible combinations of Sraffa effects.

The demonstration and visualization of these results is presented in an open and disaggregated model of prices of production. The functional distribution of income between wages, rents, and profits is not specified. The approach illustrated in these blog posts, in some sense, provides an even more open model. Prices of production are consistent with the smooth reproduction of a capitalist economy. In specifying prices of production, technology, relative rates of profits among industries, and requirements for use are frozen. Fluke cases are found, on the other hand, by perturbing parameters that specify these givens for prices of production.

Whatever practical conclusions can be drawn from this widening of the horizon remain on a high level of abstraction. Characteristics of the conflict over the functional distribution of income between wages and profits can depend on struggle within the class of capitalists. Landlords, in as much as they their interests are reflected in the existence and size of the rent of specific types of land, are also affected by the conflicts between workers and capitalists and within the class of capitalists. Variations in technology, in causes of persistent differences in the rate of profits among industries, and in the requirements for can change these characteristics of these conflicts.

## 4 comments:

Do you have a theory, or some suggested reading, on the generalisation of these perverse results? Given that all these are outcomes of solution of sets of linear equations (or non linear as Steedman and Opocher has shown) there must exist a topological or algebraic way to see what are the necessary (and why not the sufficient) conditions for the emergence of such "perversions"?

DM

If you want to move from lots of linear algebras describing special cases but with lots of suggestions of higher regularities to a more general conception, the kind of trendy-but-essentially simple maths I would guess is applicable is tropical geometry. Cf. e.g., https://www.imsa.miami.edu/_assets/pdf/imsa-inaugural-talk-lupercio.pdf

I am going to talk about fluke switch points, and not so-called 'perverse' results.

To find switch points, one solves a polynomial equation. My fluke cases impose some other conditions.

Algebraic geometry considers how some structures in some space of coefficients relate to the solutions of systems of polynomial equations.

I do not understand algebraic geometry and have never heard of tropical geometry before. If I want to draw on bifurcation theory, I must connect up my results to some sort of (cross-dual?) dynamics.

I will probably continue to present what I am doing as exploratory mathematics.

Kurz and Salvadori's 1995 textbook is probably still the most comprehensive presentation of post-Sraffian price theory. I like Opocher and Steedman's book.

Tropical geometry is a geometry that arises from tropical algebra, which is a kind of super-tractable theory of polynomials where instead of the ring being based on addition and multiplication, it is based on the maximum operator and addition.

Tropical algebras arise naturally from families of linear algebras if the linear algebras are well-behaved: if the family of linear algebras form an ideal, then the ideal should have a boundary and it is likely that that boundary can be described using a tropical polynomial. The geometry is like algebraic geometry in that we are interested in solutions to equations involving polynomials, but where traditional algebraic geometry gives rise to deeply inaccessible research efforts like the Langlands program, equations involving tropical polynomials are really easy to work with.

While given what I gather to be you grasp of linear algebra I don't think it would take much time for you to get the basics of tropical algebra, the point of my comment was not to recommend that you do that. Rather, it's that I think the results you are putting together are getting to the point that it is possible that they might be a jumping off point for some quite appealing applied mathematics. The point about simplicity is that if a mathematician were willing to take this idea up, I am guessing that you would not find it so hard to follow what they were doing. I don't think the distance from your exploratory mathematics is far from what

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