A Double-Fluke Switch Point For A Triple-Switching Example

Figure 1: The Wage Frontier for a Double-Fluke Switch Point

This post is an expansion of a previous one. That post defines the technology and the price system for the three techniques conprising the technology. In Alpha, labor and corn inputs are used to produce corn. Beta and Gamma are more roundabout. In each, Labor and corn are first used to make a machine that lasts two years and can be used to produce corn each year. In Beta, the machine is discarded after being operated one year. The machine is operated for its full life of two years in Gamma. The technology varies over time. One pararemter specifies the decrease in coefficients of production for Alpha.

The solutions of the price system for a technique yields a wage curve. Figure 1 plots the wage curves for the three techniques for a selected time and rates of decrease of the coefficients of production. The cost-minimizing technique at a given rate of profits is the technique with the maximum wage at that rate. In the illustrated example, Gamma is always cost-minimizing, and Alpha is also cost-minimizing at a rate of profits of zero. The switch point between Alpha and Gamma is a fluke in two ways. It is on the wage axis, and the Alpha and Gamma wage curves are tangent at the switch point.

The outer frontier has certain properties in models of pure fixed capital. Wage curves are downward-sloping on the frontier. A maximum wage corresponds to a rate of profits of zero, and a maximum rate of profits corresponds to a wage of zero. The intersection of a wage curve with the wage axis is the output of numeraire per worker for the technique in a stationary state. In a stationary state, the net output is consumed. A higher value of capital per worker goes hand-in-hand with a higher output of corn per worker.

The choice of technique between the Alpha and Gamma techniques, at a given rate of profits, can also be analyzed by examining whether of not extra profits can be obtained in operating the only process comprising the Alpha technique when prices of production for Gamma prevail. The cost of the seed corn and the services of labor can be summed for this process, when operated at unit level. The cost of the seed corn includes a charge for the given rate of profits. Extra profits are the difference between revenues and this sum. Figure 2 illustrates that extra profits are always negative for Alpha at this point in the parameter space, except for the switch point at a zero rate of profits. In the remainder of these posts, this method is used to analyze the choice of technique, since the graph in Figure 2 is more visually compelling than the wage frontier in Figure 1. The Beta technique is never cost-minimizing for the parameters examined here.

Figure 2: Extra Profits for Gamma Prices for the Double-Fluke Switch Point

The double-fluke case examined so far occurs at the point in the parameter space highlighted in Figure 3. Each of the two fluke properties of the double-fluke case correspond to a locus in the parameter space. The boundary between regions 2 and 3 is for cases in which a switch point is on the wage axis. It is hard to distinguish this locus by eye in Figure 3 from the boundary between regions 3 and 4, which is for fluke switch points in which the wage curves for Alpha and Gamma are tangent. A third fluke property, a switch point on the axis for the rate of profits, corresponds to the boundary between regions 1 and 2.

Fluke properties of switch points partition the parameter space into regions. The analysis of the choice of technique is qualitatively invariant in each region. Table 1 summarizes the choice of technique in each region. Techniques are listed in order of an increasing rate of profits. One can check that the variation of the order of cost-minimizing techniques among regions is consistent with the fluke properties of the boundaries between regions.

Figure 3: Structural Dynamics around the Double-Fluke Switch Point

Table 1: Cost-Minimizing Technique byRegion

Region	Cost-Minimizing Technique	Notes
1	Alpha	No switch point.
2	Alpha, Gamma	Around the switch point, a lower rate of profits is associated with a LESS round-about technique and greater output per worker.
3	Gamma, Alpha, Gamma	Around the second switch point, a lower rate of profits is associated with a LESS round-about technique and LOWER output per worker.
4	Gamma	No switch point.

So far, the partitions in the parameter space have not outlined a region in which triple-switching occurs. Region 3 is a region in which reswitching occurs.