An Algorithm Trace For The Truncation Of Fixed Capital

1.0 Introduction

This post revisits my example of the recurrence of truncation without reswitching. In this example, the choice of technique consists of deciding on the economic life of a machine in each industry. I present an application of an algorithm to find the cost-minimizing technique, given the rate of profits. The algorithm needs more elaboration. A trace of the algorithm is a dynamic path through the space of techniques.

2.0 Technology and Techniques

I repeat the parameters that define the example in this section.

Tables 1 and 2 show the inputs and outputs for each process known to the managers of firms. For example, the inputs for the first process, at a unit level of operation, consist of 1/10 person-years, 1/16 bushels corn, and one new machine. The outputs, available after a year, are two new machines and one machine a year older.

Table 1: Inputs for The Technology

Input	Industry
	Machine		Corn
	I	II	III	IV
Labor	1/10	8	43/40	1
Corn	1/16	3/20	1/8	53/200
New Machines	1	0	1	0
One-Year Old Machines (1st type)	0	1	0	0
One-Year Old Machines (2nd type)	0	0	0	1

Table 2: Outputs for The Technology

Output	Industry
	Machine		Corn
	I	II	III	IV
Corn	0	0	1	14/25
New Machines	2	5/2	0	0
One-Year Old Machines (1st type)	1	0	0	0
One-Year Old Machines (2nd type)	0	0	1	0

With this specification of the technology, the economic life of the machine must be chosen in each industry. Table 3 lists the available techniques. The machine is truncated in both industries in the Alpha technique. The machine is operated for its full physical life in both industries in the Delta technique. In Beta and Gamma, the machine is truncated in one industry and operated for its full physical life in the other.

Table 3: Specification of Techniques

Technique	Processes	Notes
Alpha	I, III	Machines truncated in both industries.
Beta	I, II, III	Machines truncated in machine-production.
Gamma	I, III, IV	Machines operated at full physical life in both industries.
Delta	I, II, III, IV	Machines truncated in corn-production.

3.0 An Algorithm for Fixed Capital

I now present a hand-waving, incomplete specification of an algorithm for the choice of technique. This algorithm is supposed to apply when the choice of technique consists exclusively of the choice of the economic life of a machine in various industries.

1. Solve price system, given the rate of profits, for each technique.
2. Identify technique in which machines are operated for two years (longest in example).
• Beta and DELTA in the machine industry
• Gamma and DELTA in the corn industry
3. Find price of old machine in each industry. If it is negative, truncate to longest time in which it is first negative.
4. If a machine is truncated in any industry, repeat previous step.
5. For COST-MINIMIZING technique, prices of old machines are non-negative in all industries.

4.0 Traces

Which order should industries be considered? This is one way the above specification is incomplete. Maybe I should say this is a non-deterministic algorithm. Anyways, Table 4 shows the application of this algorithm starting with the first industry in the example.

Table 4: The Algorithm, Starting with the Machine Industry

Calculate the price of an old machine in the machine industry with Delta prices.
Price negative for 0 ≤ r < 71.2 percent		Price positive for 71.2 percent < r ≤ Rδ
Truncate to Gamma		Keep Delta
Calculate the price of an old machine in the corn industry with Gamma prices.		Calculate the price of an old machine in the corn industry with Delta prices.
Price negative for 0 ≤ r < 70.2 percent	Price positive for 70.2 < r < 71.2 percent	Price positive for 71.2 < r < 87.5 percent	Price negative for 87.5 percent < r < Rδ
Truncate to Alpha	Keep Gamma	Keep Delta	Truncate to Beta
Calculate the price of an old machine in the machine industry with Beta prices.			Calculate the price of an old machine in the machine industry with Beta prices.
Price negative for 0 ≤ r < 70.2 percent			Price positive for 87.5 < r ≤ Rδ
Keep Alpha			Keep Beta

Perhaps the termination criterion for the algorithm should include that a longer economic life of machines has been considered in each industry. In the range of profits in which Alpha is cost-minimizing, I consider extending the economic life of the machine in the corn industry, for the start of the last three rows. These steps extend the algorithm in section 3. Are these steps necessary? When I find a negative price for an old machine in such an extension, can I stop? Or should I, in other examples, continue consider extensions up to the physical life of the machine? I know that truncation can jump from three years, for example, to one year.

Table 5 shows the application of the algorithm starting with the second industry in the example. These two tables illustrate that it does not matter which industry is considered first. I suppose this algorithm, like Christian Bidard's market algorithm, could be distributed across industries, with steps being executed in parallel. I think that if somebody was going to elaborate on this claim, they should consider a specification of market algorithms in a language designed for parallel processing, such as Tony Hoare's Communicating Sequential Processes.

Table 5: The Algorithm, Starting with the Corn Industry

Calculate the price of an old machine in the corn industry with Delta prices.
Price positive for 0 ≤ r < 87.5 percent			Price negative for 87.5 percent < r ≤ Rδ
Keep Delta			Truncate to Beta
Calculate the price of an old machine in the machine industry with Delta prices.			Calculate the price of an old machine in the machine industry with Beta prices.
Price negative for 0 ≤ r < 71.2 percent		Price positive for 71.2 < r < 87.5 percent	Price positive for 87.5 < r ≤ Rδ
Truncate to Gamma		Keep Delta	Keep Beta
Calculate the price of an old machine in the machine industry with Gamma prices.
Price negative for 0 ≤ r < 70.2 percent	Price positive for 70.2 < r < 71.2 percent
Truncate to Alpha	Keep Gamma
Calculate the price of an old machine in the machine industry with Beta prices.
Price negative for 0 ≤ r < 70.2 percent
Keep Alpha

5.0 Conclusion

How should the algorithm be modified for a rate of profits towards the maximum? Can a proof be found that the convergence of the algorithm does not depend on the order in which industries are considered? Once a machine is truncated, is it true, the extension of the economic life a machine need never be considered in any industry?