More On Truncation Without Reswitching

Figure 1: Partitions For A Parameter Space

This post extends this and this. I am considering perturbations of coefficients of production in a numerical example of the recurrence of truncation without the reswitching of techniques.

The example consists of two industries that produce new machines and corn, respectively. The physical life of a machine is two years in each industry. Corn acts as circulating capital, as well as the consumption good.

As the result of technical change, coefficients of production vary. No variation in labor coefficients or of the output matrix are considered here so as to retain forward substitution of labor in the corn industry at the switch point between Alpha and Gamma and reverse labor substitution at the switch point between Beta and Delta. Accordingly, I consider permutations of the coefficients of production in the first row of the input matrix. These coefficients of production specify inputs of circulating capital for each process, when operated at unit level.

Previously, I considered perturbations of the pair of coefficients that specify the inputs of corn needed for each machine when the machine is run in the second year of its life. Today, I consider perturbations in a_1,1 and a_1,3, the coefficients that specify the inputs of corn needed for each machine when the machine is run in the first year of its life.

Figure 1, at the top of this post, partitions a part of the parameter space. In drawing this figure, a_1,2 = 0.635 and a_1,4 = 0.319. Each partition corresponds to a fluke switch point. For the values of a_1,2 and a_1,4 in Table 1 in this post, the parameters corresponding to the fluke switch point in which the four wages curves intersect on the axis for the rate of profits is an economically meaningless point in the second quadrant.

A switch point is a fluke if it is a knife edge case in which almost all perturbations of model parameters destroy its defining properties. Four of the five partitions correspond to a switch point on the axis for the rate of profits. With four techniques, switch points between six pairs are possible. A switch point between Alpha and Delta can only occur as an intersection of all four curves. The same goes for a switch point between Beta and Gamma. These two pairs of techniques correspond to the partition for the switch point at which all four wage curves intersect.

The sequence of techniques along the wage frontier is invariant in each numbered region in the slice of the parameter space in Figure 1. Table 1 lists the cost-minimizing techniques in each region, in order of an increasing rate of profits. Other partitions, depicted in Figure 2, exist up and to the right. One is for a switch point between Alpha and Gamma on the wage axis, and another is for the switch point between Gamma and Delta on the wage axis. Three new regions are associated with these partitions. Gamma is cost-minimizing, whatever the distribution of income, for a high enough value of a_1,3, not exceeding unity, in the upper left. Delta is cost-minimizing alone in a region to the northeast. Gamma and Delta are cost-minimizing, in that order for an increasing rate of profits, in a region bounded below by region 3 and also by these two additional regions.

Table 1: Overview of Regions

Region	Techniques	Notes
1	Alpha	No switch point.
2	Alpha, Gamma	Lower rate of profits associated with truncation in corn industry, greater output per worker.
3	Alpha, Gamma, Delta	Lower rate of profits associated with truncation, greater output per worker.
4	Alpha, Gamma, Delta, Beta	Recurrence of truncation in corn industry.
5	Alpha, Beta	Lower rate of profits associated with truncation in machine industry, greater output per worker.
6	Gamma	No switch point.
7	Gamma, Delta	Lower rate of profits associated with truncation in corn industry, greater output per worker.
8	Delta	No switch point.

Figure 2: An Expanded View of Partitions For A Parameter Space

Suppose technical change lowers corn inputs for the first year a machine is operated, in either industry, while leaving all other coefficients of production unchanged. Some trajectories going roughly from the northeast to the southwest in Figures 1 and 2 cross through region 4. Under this specific form of technical progress, the recurrence of truncation can arise and then disappear in secular time. But it need not.