The Roots of a Cubic Polynomial Defining Switch Points |

I have a draft paper up at SSRN. The abstract:

This paper illustrates, through a numerical example of reswitching under oligopoly, the existence of implications from the Cambridge Capital Controversy for the theory of industrial organization. Oligopoly is modeled by given and persistent ratios in rates of profits among industries, as expressed in a system of equations for prices of production. The numerical example illustrates that this model of oligopoly is a pertubation of free competition. Some comparisons and contrasts are drawn to a model of free competition.

In some sense, this paper shows a somewhat more comprehensive description of value through exogenous distribution than in Sraffa's book. The model can depict capitalists as squabbling over the division of the surplus that their class gets, as well as their struggle against the workers. I'd like to see an example of reswitching or capital reversing in this model, with all (price and real) Wicksell effects as negative in the example in the special case of free competition. I do not see why one cannot arise. Such an example would suggest that "perverse" examples can obtain empirically, even if they are not found in an analysis that presumes one common rate of profits among all industries.

The graph at the top of this post does not appear in the paper. In the model, the ratios of rates of profits among industries are given parameters. A cubic polynomial is defined for a given set of such ratios. Non-negative, real zeroes of that polynomial below a certain maximum define a scale factor for switch points. The location of the zeros varies with the ratios. I happen to be able to solve for the zeros. They are shown in the graph above.