Recurrence Of Truncation Without Reswitching: Another Example

Figure 1: Demand for Labor in the Corn Industry

1.0 Introduction

I have previously worked through an example of the recurrence of truncation with neither the reswitching of techniques nor capital-reversing. This post reports the perturbation of several coefficients of production in that example: a_0,1, a_0,2, a_1,2, and a_1,3.

This modification of the example lowers the rate of profits at which the recurrence of truncation occurs, from approximately 87.5 percent to 67.6 percent. That rate of profits may still be taken to be too high to be observed empirically. Also, in practice, you may want the coefficients of production not to vary too much among processes. The ratio of a_0,2 to a_0,1 is now 7.5, much lower than before. Presumably, with many more produced commodities and machines with physical lives of more production cycles, the recurrence of truncation might appear at a lower rate of profits.

2.0 Technology

Two industries exist in the example. One industry produces machines, and the other industry produces corn. Corn is a consumption good, the good for circulating capital, and the numeraire. Machines are fixed capital. Each machine has a physical life of two years. Old machines cannot be transferred between industries. I assume constant returns to scale (CRS) and the free disposal of old machines. Labor is advanced and paid out of the surplus of corn.

Tables 1 and 2 show the inputs and outputs for each process known to the managers of firms. For example, the inputs, at a unit level of operation, consist of 4/5 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	4/5	6	43/40	1
Corn	1/16	1/10	13/100	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
Alpha	I, III
Beta	I, II, III
Gamma	I, III, IV
Delta	I, II, III, IV

3,0 The Choice of Technique

The economic life of a machine is chosen to minimize cost. A system of equations for prices is associated with each technique. This system can be solved. In the solution, the wage is a function of the rate of profit. Each price of a produced commodity is also a function of the rate of profits.

Figure 2 shows the wage curves, for the four techniques in the example. The cost-minimizing technique at each wage or rate of profits is the technique with its wage curve on the outer frontier. The cost-minimizing techniques are indicated on the figure.

Figure 2: Wage Curves In The Example

At any rate, the cost-minimizing techniques, in order of an increasing rate of profits, are Alpha, Gamma, Delta, and Beta. Each pair of techniques at a switch point on the frontier differs in one process. A switch point in which the economic life of a machine differs in both industries would be a fluke case. No fluke switch points exist in this example, without perturbing some coefficients of production.

Table 4 summarizes the results for this numerical example. The bounds on the ranges of the rates of profits are approximate.

Table 4: Cost-Minimizing Techniques

Range	Technique	Truncation
0 ≤ r ≤ 46.3%	Alpha	Machines truncated in both industries.
46.3% ≤ r ≤ 49.6%	Gamma	Machines truncated in machine-production.
49.6% ≤ r ≤ 67.6%	Delta	Machines operated at full physical life in both industries.
67.6% ≤ r ≤ 129.4%	Beta	Machines truncated in corn-production.

At any rate, the machine is truncated in corn-production when both the Alpha and the Beta technique are cost-minimizing. The truncation of the machine in corn-production recurs, being part of the cost-minimizing technique at extremes of low and high rates of profits. This is not, however, an example of the reswitching of techniques.

Around the switch point between Beta and Delta, an increased wage is associated with increased labor inputs in the corn industry, per bushel corn produced in that industry (Figure 1). At each point on the graph, the production system as a whole is in equilibrium, in the sense that no extra profits can be obtained. Both the rate of profits and the price of a new machine vary with the wage. The switch points between Alpha and Gamma and between Beta and Delta appear as horizontal line segments on this graph. Since the economic life of a machine in corn production does not vary at the switch point between Gamma and Delta, that switch point is only a point on the graph. The switch point between Beta and Delta is an example of the reverse-substitution of labor. The example illustrates that the reverse substitution of labor and the recurrence of truncation are independent of the reswitching of techniques and of capital-reversing.

4.0 Conclusion

The CCC developed a critique of aggregate production functions and aggregate neoclassical theory. But the critique also extends to microeconomics. The recurrence of truncation, like process recurrence (Steedman 2002), re-iterates that the concept of vertically-integrated industries provides a coherent framework for partial analysis. Both process recurrence and this example of the recurrence of truncation challenge claims that, under ideal conditions, the long-period demand for labor from a particular industry is a non-increasing function of the real wage. These demonstrations about industry-specific analyses do not rely on either the reswitching of techniques or capital-reversing.