Update To A Start On A Catalog Of Switch Point Patterns Of High Co-Dimension

I have been looking at patterns of switch points. A pattern is a configuration of switch points helpful for perturbation analysis for the choice of technique. I am curious how the switch points and the wage curves along the wage frontier can alter with parameters, in a model of the production of commodities. Such a parameter can be a coefficient of production; time, where a number of parameters are functions of time; or the markup in an industry or a number of industries. A normal form exists for each pattern. The normal form describes how the techniques and switch points along the frontier vary with a selected parameter value. Each pattern is defined by the equality of wage curves at a switch point and one or more additional conditions. The co-dimension of a pattern is the number of additional conditions.

I claim that local patterns of co-dimension one, with a switch point at a non-negative, feasible rate of profits can be described by four normal forms. I have defined these patterns as a pattern over the axis for the rate of profits, a pattern across the wage axis, a three-technique pattern, and a reswitching pattern. This post is an update, and continues to examine global patterns, local patterns with a co-dimension higher than unity, and sequences of local patterns. Some examples are:

• A switch point that is simultaneously a pattern across the wage axis and a reswitching pattern (a case of a real Wicksell effect of zero). This illustrates a pattern of co-dimension two.
• A reswitching example with one switch point being a pattern across the wage axis (another case of a real Wicksell effect of zero). This is a global pattern.
• An example with a pattern across the wage axis and a pattern over the axis for the rate of profits. This is a global pattern.
• A pattern like the above, but with both switch points being defined by intersections of wage curves for the same two techniques. This is a global pattern.
• Two switch points, with both being reswitching patterns, can be found from a partition of a parameter space where two loci for reswitching patterns intersect. This gestures towards a global pattern.
• A pattern across the point where the rate of profits is negative one hundred percent, combined with a switch point, for the same techniques, with a positive rate of profits (of interest for the reverse substitution of labor). This is a global pattern.
• An example where every point on the frontier is a switch point. This is a global pattern of an uncountably infinite co-dimension.
• Speculation on three sequences of patterns of co-dimension one that result in one technique replacing another, in an intermediate range of the rate of profits, along the wage frontier.
• A switch point for a four-technique pattern (due to Salvadori and Steedman). This is a local pattern of co-dimension two.
• Further analysis of the above example.
• An example of a four-technique pattern in a model with three produced commodities. This local pattern of co-dimension two results in one technique replacing another, in an intermediate range of the rate of profits, along the wage frontier.
• Further analysis of the above example. Two normal forms are identified for four-technique patterns.

The above list is not complete. More types of fluke switch points exist. Some, like the examples of a real Wicksell effect of zero, I thought, should be of interest for themselves to economists. Others show examples of parameters where the appearance of the wage frontier, at least, changes with perturbations of the parameters. I have used these patterns to tell stories about how technical change or a change in markups (that is, structural economic dynamics) can result in reswitching, capital reversing, or the reverse substitution of labor appearing on or disappearing from the wage frontier.

I would like to see that in at least some cases, short run dynamics changes qualitatively with such perturbations. But this seems to be beyond my capabilities.