Recurrence Of Truncation In A Perturbation Analysis

Figure 1: Variation of Choice Of Technique with a Coefficient of Production

This post continues the analysis of this example. The coefficients of production and the techniques are the same as in the linked post, except here I consider the results of varying a_{1, 2}, the amount of corn needed as circulating capital in operating Process II at unit level. Figure 1 above shows how the choice of technique varies with this parameter. This is a two-sector model, in which new machines and corn are produced in both sectors. Corn acts as circulating capital, as the sole consumption good, and as the numeraire. Machines act as fixed capital. Managers of firms have the ability to run machines for two years in both sectors, but old machines cannot be transferred between sectors.

Table 1: Variation in the Choice of Technique

1	0 ≤ r ≤ r1	Operation of machine truncated in both sectors.
	r1 ≤ r ≤ rβ	Machine operated for 2 years in machine sector. Truncated after 1 year in corn sector.
2	0 ≤ r ≤ r1	Operation of machine truncated in both sectors.
	r1 ≤ r ≤ r2	Operation of machine truncated after 1 year in machine sector. Machine operated for 2 years in corn sector.
	r2 ≤ r ≤ r3	Machine operated for two years in both sectors.
	r3 ≤ r ≤ rβ	Machine operated for 2 years in machine sector. Truncated after 1 year in corn sector.
3	0 ≤ r ≤ r1	Operation of machine truncated in both sectors.
	r1 ≤ r ≤ r2	Operation of machine truncated after 1 year in machine sector. Machine operated for 2 years in corn sector.
	r2 ≤ r ≤ rδ	Machine operated for two years in both sectors.
4	0 ≤ r ≤ r1	Operation of machine truncated in both sectors.
	r1 ≤ r ≤ rγ	Operation of machine truncated after 1 year in machine sector. Machine operated for 2 years in corn sector.

One can tell a tale with this example by reading Figure 1 from right to left. Initially, managers of firms do not find it profitable to run a machine for more than one year in the machine sector. At a high rate of profits (or a low wage), they want to run the machine for two years in the corn sector. Meanwhile, the engineers are figuring out how to use less circulating capital with old machines in the sector manufacturing new machines. Eventually, as at the start, firms run machines in both sectors for one year for low rates of profits. But which sector they want to run machines for two years in at high rates of profits is reversed; at the end they run machines for two years only in the machine sector at high rates of profits. The above figure and table show this change coming about.

Table 2: Summary of Results

1	Larger rate of profits associated with longer economic life of machine in machine sector
2	Larger rate of profits associated with longer economic life of machine in machine sector; Recurrence of truncation in corn sector
3	Larger rate of profits associated with longer economic life of machine in both sectors
4	Larger rate of profits associated with longer economic life of machine in corn sector

The example can be used to highlight the falsity of outdated, archaic intuition associated with neoclassical and Austrian-school price theory. One might think that a desire of individuals to save more would be associated with an increased supply of capital. And this would drive the interest rate down. A lower interest rate would supposedly induce firms to adopt more capital-intensive techniques and to run machinery longer. The adoption of more capital-intensive techniques would, in turn, lead to greater output per worker. Around each switch point in the example, a lower interest rate is indeed associated with the adoption of a technique which provides greater output per head. (This property does not generalize, as shown by examples of capital-reversing, also known as positive real Wicksell effects.) But, in the example, a lower interest rate is associated with the truncation of the economic life of machines, except in the corn sector in Region 2. In that region, a lower interest rate is first associated with an increase of the economic life of the machine and then, for an even lower interest rate, a truncation of its life in the corn sector. The economic life of the machine does not even bear a monotonic relationship with the interest rate.

Notice that I am not arguing about aggregate measures of capital or the so-called average period of production. An analysis of prices of production shows that out-dated neoclassical and Austrian-school economists just have a faulty understanding of microeconomics, of how prices work.

I guess this example is something different than the recurrence of a process, given the specific manner in which fixed capital is involved. If you have thoroughly absorbed post Sraffian price theory, the recurrence of truncation is, I guess, no surprise. My contribution is in visualizing this possibility.