Perturbation Of A Model Of The Recurrence Of Truncation

Figure 1: A Partition of Part of the Parameter Space

1.0 Introduction

This post is a continuation of the example in a previous post. That example is of the recurrence of truncation without reswitching. I here consider perturbations of selected coefficients of production and of relative markups in the two industries.

2.0 Perturbation of Efficiency of Circulating Capital for Old Machines

The first industry in the example produces machines, and the second produces corn. Machines constitute fixed capital, and corn functions as circulating capital, as well as a consumption good.

Two processes are available in each industry. The second process uses an old machine to produce the output of that industry, whether a new machine or corn. Old machines are specific to the industry in which they were (jointly) produced. The choice of technique, in this model of pure fixed capital, is equivalent to choosing to truncate the economic life of a machine in either industry. The four possible techniques are defined, along with the technology, in the previous post.

Consider perturbations of a_1,2 and a_1,4. The first coefficient of production is the amount of corn input, as circulating capital, needed to operate an old machine in the machine industry. The second coefficient of production is the input of corn needed to operate an old machine in the corn industry.

Figure 1, at the top of this post depicts a partition of the resulting parameter space. The variation of the choice of technique with distribution is invariant in each numbered region. Table 1 lists the cost-minimizing techniques, in order of an increasing rate of profits in each region.

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.

The diagram yields the following results:

• Region 1: If old machines are inefficient enough, then the economic life of machine is one year.
• Region 5: With improvement in the efficiency of old machines in the machine industry, machines are operated for two years in the machine industry at large rates of profits.
• Region 2: With improvement in the efficiency of old machines in the corn industry, machines are operated for two years in the corn industry at large rates of profits
• Region 4: Recurrence of the truncation of machines in the corn industry occurs at a specific range of these coefficient of production.
• Region 3: With sufficient improvement in the efficiency of old machines, no possibility arises of the economic life of machine being operated for two years only in the machine industry, whatever the distribution of income.

All these regions are around a quintuple fluke switch point. The partition between regions 1 and 5 occurs for parameters for which managers of firms are indifferent, when the wage is zero, about the economic life of a machine in producing new machines. The partition between regions 1 and 2 occurs when managers of firms are indifferent, also at a wage of zero, about the economic life of a machine in producing corn. The intersection of these two partitions must also be an intersection of the other three partitions.

3.0 Perturbation of Relative Markups

Now suppose the technology is fixed, as in the post. Let s₁ r be the rate of profits in the machine industry, and s₂ r the rate of profits in the corn industry. As a normalization condition, I assume the sum of the relative markups is unity:

s₁ + s₂ = 1

Figure 2 displays the effects on the choice of technique of perturbations of persistent relative markups.

Figure 2: Effects on The Choice of Technique of Perturbations of Relative Markups

High enough market power for the corn industry, as compared to market power in the machine industry, can eliminate the possibility of extending the economic life of the the machine in the corn industry. This is seen in region 5, to the left on the graph. Persistent high enough market power for the machine industry, can also eliminate the recurrence of the truncation of machines in the corn industry. This is seen in region 3 to the right on the above graph.

4.0 Conclusion

The choice of technique varies with variations in distribution between wages and profits. Both technical progress and changes in market power can have similar effects, in the large. In this example, both can bring about or eliminate the recurrence of the truncation of the economic life of the machine in one industry. Technical progress, however, has the benefit of increasing productivity.