Bifurcation Analysis in a Model of Oligopoly

Figure 1: Bifurcation Diagram

I have presented a model of prices of production in which the the rate of profits differs among industries. Such persistent differential rates of profits may be maintained because of perceptions by investors of different levels of risk among industries. Or they may reflect the ability of firms to maintain barriers to entry in different industries. In the latter case, the model is one of oligopoly.

This post is based on a specific numeric example for technology, namely, this one, in which labor and two commodities are used in the production of the same commodities. I am not going to reset out the model here. But I want to be able to refer to some notation. Managers know of two processes for producing iron and one process for producing corn. Each process is specified by three coefficients of production. Hence, nine parameters specify the technology, and there is a choice between two techniques. In the model:

• The rate of profits in the iron industry is rs₁.
• The rate of profits in the corn industry is rs₂.

I call r the scale factor for the rates of profits. s₁ is the markup for the rate rate of profits in the iron industry. And s₂ is the markup for the rate of profits in the corn industry. So, with the two markups for the rates of profits, 11 parameters specify the model.

I suppose one could look at work by Edith Penrose, Michal Kalecki, Joseh Steindl, Paolo Sylos Labini, Alfred Eichner, or Robin Marris for a more concrete understanding of markups.

Anyways, a wage curve is associated with each technique. And that wage curve results in the wage being specified, in the system of equations for prices of production, given an exogenous specification of the scale factor for the rates of profits. Alternatively, the scale factor can be found, given the wage. Points in common (intersections) on the wage curves for the two techniques are switch points.

Depending on parameter values for the markups on the rates of profits, the example can have no, one, or two switch points. In the last case, the model is one of the reswitching of techniques.

A bifurcation diagram partitions the parameter space into regions where the model solutions, throughout a region, are topologically equivalent, in some sense. Theoretically, a bifurcation diagram for the example should be drawn in an eleven-dimensional space. I, however, take the technology as given and only vary the markups. Figure 1, is the resulting bifurcation diagram.

The model exhibits a certain invariance, manifested in the bifurcation diagram by the straight lines through the origin. Suppose each markup for the rates of profits were, say, doubled. Then, if the scale factor for the rates of profits were halved, the rates of profits in each industry would be unchanged. The wage and prices of production would also be unchanged.

So only the ratio between the markups matter for the model solution. In some sense, the two parameters for the markups can be reduced to one, the ratio between the rates of profits in the two industries. And this ratio is constant for each straight line in the bifurcation diagram. The reciprocal of the slopes of the lines labeled 2 and 4 in Figure 1 are approximately 0.392 and 0.938, respectively. These values are marked along the abscissa in the figure at the top of this post.

In the bifurcation diagram in Figure 1, I have numbered the regions and the loci constituting the boundaries between them. In a bifurcation diagram, one would like to know what a typical solution looks like in each region and how bifurcations occur. The point in this example is to understand changes in the relationships between the wage curves for the two techniques. And the wage curves for the techniques for the numbered regions and lines in Figure 1 look like (are topologically equivalent to) the corresponding numbered graphs in Figure 2 in this post

The model of oligopoly being analyzed here is open, insofar as the determinants of the functional distribution of income, of stable relative rates of profits among industries, and of the long run rate of growth have not been specified. Only comparisons of long run positions are referred to in talking about variations, in the solution to a model of prices of production, with variations in model parameters. That is, no claims are being made about transitions to long period equilibria. Nevertheless, the implications of the results in this paper for short period models, whether ones of classical gravitational processes, cross dual dynamics, intertemporal equilibria, or temporary equilibria, are well worth thinking about.

Mainstream economists frequently produce more complicated models, with conjectural variations, or game theory, or whatever, of firms operating in non-competitive markets. And they seem to think that models of competitive markets are more intuitive, with simple supply and demand properties and certain desirable properties. I think the Cambridge Capital Controversy raised fatal objections to this view long ago. Reswitching and capital reversing show that equilibrium prices are not scarcity indices, and the logic of comparisons of equilibrium positions, in competitive conditions does not conform to the principle of substitution. In the model of prices of production discussed here, there is a certain continuity between imperfections in competition and the case of free competition. The kind of dichotomy that I understand to exist in mainstream microeconomics just doesn't exist here.