A Fixed Capital System That Is Or Is Not Interlocked

I have defined patterns of switch points in considering perturbations of examples of the choice of technique. For example, I have defined three-technique and four-technique patterns. An obvious extension is to consider how these patterns arise in models of joint production. A simplification is to only consider models of fixed capital without superimposed joint production.

This post lays out an example in which, maybe, some parameter values can lead to a three-technique pattern. I am trying to consider whether I want to allow it to be an interlocked system. The question arises when I lay out a simple example in which a machine can be used for one or two years.

Accordingly, consider an example with the coefficients of production in Table 1. Corn and new machines are finished goods. An old machine is an intermediate good. Intermediate goods cannot be sold for consumption. Processes I and II constitute the machine-producing or machine sector. Processes III and IV are in the corn sector. In each sector, there is one and only one primary process, in which the only inputs are finished goods. Processes I and III are the primary processes in the two sectors. Processes II and IV are secondary processes.

Table 1: Coefficients of Production for The Technology

Input	Process
	(I)	(II)	(III)	(IV)
Labor	a0,1	a0,2	a0,3	a0,4
Corn	a1,1	a1,2	a1,3	a1,4
New Machines	1	0	1	0
Old Machines	0	1	0	1
Outputs
Corn	0	0	b1,3	b1,4
New Machines	b2,1	b2,2	0	0
Old Machines	1	0	1	0

I make the usual assumption, inappropriate for environmental economics, of free disposal. A machine can be discarded without cost after used for only one year; managers of firms can choose not to operate the secondary processes in either sector. I also make the assumption that old machines cannot be transferred between sectors. An old machine jointly produced by the first process can be either discarded or used to operate process II. It cannot be used as an input in process IV. With these assumptions, this is a pure fixed capital system without superimposed joint production. An analysis of the choice of technique must consider the four techniques in Table 2. The cost-minimizing technique can be found by constructing the wage frontier. Demand does not matter except inasmuch as coefficients of production are affected by the scale at which processes are operated.

Table 2: Techniques

Technique	Processes
Alpha	I, III
Beta	I, II, III
Gamma	I, III, IV
Delta	I, II, III, IV

Table 3: Variables with Corn as Numeraire

Variable	Definition
w	Wage, in units of bushels per person-year
r	The rate of profits
p1	Price of a new machine, in units of bushels per machine
p2	Price of an old machine in the machine sector, in units of bushels per machine
p3	Price of an old machine in the corn sector, in units of bushels per machine

Suppose, on the other hand, that an old machine can be transferred between sectors. For example, the old machine jointly produced in the first process can be used in process IV to produce corn. Its history in being used to produce new machines has no effect on its efficiency in then being used to produce corn. Then only thee techniques, Alpha, Beta, and Gamma, would exist. Delta could only arise as a switch point between Beta and Gamma. Only two types of machines would exist. And p₂ would be redefined to be the price an old machine, whatever sector it would be in. This is then an example of an interlocked system. One might think reducing the number of techniques and the number of variables would simplify the model. But a simplification of the price system is available in the non-interlocked system that I do not think is possible for the interlocked system.

I think I am convincing myself that the assumption a system cannot be interlocked is a reasonable assumption.