A Reswitching Bifurcation, Reflected

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Robert Vienneau
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Figure 1: Two Bifurcation Diagrams Horizontally Reflecting

This post continues my investigation of structural economic dynamics. I am interested in how technological progress can change the analysis of the choice of technique. I have four normal forms for how switch points can appear on or disappear from the wage frontier, as a result of changes in coefficients of production. This post concentrates on what I call a reswitching bifurcation.

Each bifurcation can be described by how wages curves look around the bifurcation before, at, and after the bifurcation. I claim that, in some sense, order does not matter. For each normal form, bifurcations can exist in either order. I have proven this, for three of the bifurcations, by constructing the normal forms in both orders. This post completes the proof by exhibiting both orders for the reswitching bifurcation.

Consider the technology illustrated in Table 1. The managers of firms know of four processes of production. And these processes exhibit Constant Returns to Scale. The column for the iron industry specifies the inputs needed to produce a ton of iron. The column for the copper industry likewise specifies the inputs needed to produce a ton of copper. Two processes are known for producing corn, and their coefficients of production are specified in the last two columns in the table. Each process is assumed to require a year to complete and uses up all of its commodity inputs. Technology is defined in terms of two parameters, u and v. u denotes the quantity of labor needed to produce a unit iron in the iron industry. v is the quantity of labor needed to produce a unit copper.

As usual, this technology presents a problem of the choice of technique. The Alpha technique consists of the iron-producing process and the corn-producing process labeled Alpha. Similarly, the Beta technique consists of the copper-producing process and the corn-producing process labeled Beta.

Consider certain specified parameter values for the coefficients of production denoting the amount of labor needed to produce one unit of iron and one unit of copper. In particular, let u be 1, and let v be 17,328/8,281. Figure 2 graphs the wage curves for the two techniques in this case.

Figure 2: Wage Curves at the Bifurcation

I call this case a reswitching bifurcation. Like all bifurcations, it is a fluke case.

Consider variations in u, from some parameter larger than its value in the above reswitching bifurcation to some lower value. In this part of the story, the value of v is assumed to be fixed at its value for the bifurcation. The right half of Figure 1, at the top of this post, illustrates this story.

For a high value of u, to the right of the right of Figure 1, the wage curve for Alpha is moved inside its location in Figure 2. The wage curves for the Alpha and Beta techniques intersect at two points. It is a reswitching example. A fall in u is illustrated by a movement to the left on the right side of Figure 1. The two switch points become closer and closer along the wage frontier. The reswitching bifurcation is illustrated by the thin vertical line in Figure 1. For any smaller value of u, the Alpha technique is cost minimizing for all feasible rates of profits or wages.

Now consider variations in v, with u fixed at the value for the bifurcation illustrated in Figure 2. Technical progress in the copper industry is illustrated by a movement to the left on the left side of Figure 1. For a high value of v, the wage curve for the Beta technique is inside the wage curve for the Alpha technique. The Alpha technique is cost-minimizing for all feasible rates of profits. As v decreases, the wage curve for the Beta technique moves outward, until it reaches the reswitching bifurcation. For smaller values of v, the example becomes, once again, a reswitching example. A second bifurcation is illustrated on the left side of Figure 1, when the switch point at the higher rate of profits moves across the axis for the wage. The labor input for copper has become so small that the Beta technique is cost-minimizing for any sufficiently large enough wage and small rate of profits.

The bifurcation depends on a certain relative configuration of wage curves, in which one is tangent to the other at a switch point. Whether technical progress around the bifurcation results in reswitching appearing or disappearing depends on which wage curve is moving outwards faster around the switch point(s). Either order is possible.