The Corn-Tractor Model With Constant 'Capital Intensity'

Figure 1: Partitioning a Part of the Parameter Space with Fluke Cases

1.0 Introduction

I have been revisiting the corn-tractor model. This analysis is confined to the case where capital-intensity does not vary between sectors for each kind of tractor. The reswitching of techniques, but not triple switching, is possible in this special case.

Capital-theoretic ‘paradoxes, such as reswitching, capital reversing, and the reverse substitution of labor, are transient in secular time when technical progress takes on certain forms. The analysis of the corn-tractor model in this post illustrates this claim. This claim provides some explanation of why empirical instances of capital-theoretic paradoxes can be hard to find empirically and complements the explantion in Schefold (2023).

2.0 Technology

Table 1 specifies the parameters for this instance of the corn-tractor model. Two industries, also known as sectors, exist. The tractor industry produces a type of tractor from inputs of labor and that type of tractor. The corn industry produces corn from inputs of labor and a given type of tractor. The tractor industry can produce multiple types of tractors. A technique is defined by the type of tractor.

Table 1: Parameters for Technology for First Example

Parameter	Type I Tractors	Type II Tractors
Tractor input per tractor	aI = 3/10	aII
Labor input per tractor	bI = 234	bII = 20
Years tractors last in tractor industry	nI = 1	nII = 2
Tractor input per bushel corn	cI = 1	cII
Labor input per bushel corn	dI = cI bI/aI	dII = cII bII/aII
Years tractors last in corn industry	mI = 1	mII = 2

I make a number of assumptions. Tractors can last for more than one year, depending on the type of tractor. For the specific numeric example, Type I tractors last for only one year. The technique operating with Type I tractors is an example of the use of circulating capital. Type II tractors are fixed capital and last for two years in both sectors. All tractors have constant efficiency and are thus operated for their full physical life. The transfer of used tractors between industries is not possible. All processes exhibit constant returns to scale (CRS). In this special case, the capital-intensity of a tractor does not vary among industries. This assumption is expressed in the definition of the parameters d_I and d_II.

The inputs of Type II tractors, per unit output of tractors and of corn, are left undefined in Table 1. Prices of production are defined for any specific values of these parameters. Technical progress in secular time is represented in the analysis in this post by decreasing values of these parameters. How the analysis of the choice of technique varies with this specific form of technical progress, in some sense, generalizes and specializes Pasinetti's structural economic dynamics.

2.0 Prices of Production

If prices of production prevail, the same rate of profits is made in both industries for the operated technique. Let r be the rate of profits, w the wage, and p₀ the price of a new tractor. Then prices of production satisfy:

p₀ a f_n(r) + w b = p₀

p₀ c f_m(r) + w d = 1

where:

f_n(r) = [r (1 + r)ⁿ]/[(1 + r)ⁿ - 1]

These equations can be solved. In the special case, the price of a new tractor does not vary with the rate of profits. This constancy is a manifestation of capital intensity not varying between industries. The wage curve for the techique is the wage expressed as a function of the rate of profits in the price system.

At a given rate of profits, the cost-minizing technique is the one with the highest wage. Two techniques are cost-minimizing at a generic switch point.

3.0 Double-Fluke Switch Points

Five double-fluke switch points exist in the range of the parameters considered here. Figure 2 shows wage curves tangent at the switch point on the wage axis. One technique is cost-minimizing for all feasible rates of profits, except at the maximum wage. There, both techniques are cost-minimizing.

Figure 2: Wage Curves Tangent on Wage Axis

Figure 3 shows another double-fluke switch point. The wage curves are tangent at the switch point on the axis for the rate of profits. One technique is cost-minimizing for all feasible rates of profits.

Figure 3: Wage Curves Tangent on Axis for Rate of Profits

Figure 4 shows a case in which one switch point is on the wage axis, and the other switch point is on the axis for the rate of profits. Like the double-fluke cases considered above, this case is a limit of the region in the parameter space at which the reswitching of techniques occurs. One technique is uniquely cost-minimizing for all feasible rates of profits, except at the extremes.

Figure 4: Switch Points on Both Axes

Figure 5 shows the fourth double-fluke case considered here. One switch point is at a rate of profits of -100 percent, and the other switch point is on the axis for the rate of profits. This case is a limit of the region in which the reverse substitution of labor occurs. In this limit, both techniques result in theemployment of the same quantity of labor per gross output in the corn industry.

Figure 5: A Pattern for the Reverse Substitution of Labor

Figure 6 shows the last double-fluke case considered here. One switch point is at a rate of profits of -100 percent, and the other switch point is on the wage axis. This case is also limit of the region in which the reverse substitution of labor occurs.

Figure 6: Another Pattern for the Reverse Substitution of Labor

4.0 A Partition of Parameter Space

Fluke switch points partition a two-dimensional slice of the parameter space (Figure 1). The sequence of cost-minimizing techniques along the wage frontier and the order of switch points are invariant in each numbered region. Region 1 exhibits the reswitching of techniques. Region 3 has the reverse substitution of labor; around the switch point a higher wage is associated with the adoption of a technique the requires more employment in the corn industry. Technical progress is a path from the northeast to the southwest in the diagram.

I would like to draw a higher-dimensional space, with one dimension for each parameter. Fluke switch points correspond to manifolds in such spaces.

The diagram for region 1 and its boundaries is generic. The same structure appears in perturbations of other examples of the reswitching of techniques and of the reswitching of the orders of fertility and of rentability in models of extensive rent.

The following properties of the partitions in Figure 1 are generic:

• The locus for the reswitching pattern ends at a point of tangency with the locus for switch points on the axis for the rate of profits.
• The other end of the locus for the reswitching pattern is a point of tangency with the locus for switch points on the wage axis.
• The loci for switch points on the axis for the rate of profits and for switch points on the wage axis intersect at a single point.
• These three loci bound a region in which reswitching occurs.

A movement from the northeast to the southwest in Figure 1 is (a specific kind of) technological progress in producing Type II tractors. Reswitching and the reverse substitution of labor can only be transient in the very long run. Eventually, the technique that produces and uses Type II tractors dominates.

The specific shapes of the partitions can differ in other models. Furthermore, the end points of the reswitching pattern may occur for parameters that are negative and beyond the bounds of economically meaningful parameter values. Furthermore, in models with more techniques, a switch point for parameter values corresponding to a double fluke case may not be on the frontier. Nevertheless, mathematical structures that arise in multiple models and multiple contexts suggest theoretic possibilities worth investigating.

5.0 Concluding Observations

This post considers structural dynamics, in which technology improves for one type of tractor in the corn-tractor model. Loci for fluke switch points partition the parameter space. The analysis illustates that reswitching, capital-reversing, and the reverse substitution of labor are transient phenomena.

Other fluke cases exist in other models. Limiting cases for triple switching, quadruple switching, and so on are examples. A case in which switch points for two wage curves exist on both the wage axis and the axis for the rate of profits, with a switch point at an intermediate rate of profits, is a fluke case for triple switching. Three wage curves intersecting at a single switch point is another fluke case not illustrated by this analysis of the corn-tractor model. Other structures in parameter spaces exist, formed by intersections of other locii for fluke cases. If parameters decrease so that, to the extent examined, one technique becomes dominant for most of the wage frontier, capital-theoretic 'paradoxes' will be difficult to find in empirical data.

Appendix: Prices and Flukes

Consider the special case in which, for a given type of tractor, capital-intensity does not vary between the two sectors and the tractor lasts for the same number of years in both sectors. The solution of the price equations is:

w = (1/d)(1 - a f_n(r))

p₀ = b/d

Consider the further special case in which a Type I lasts for one year, and a Type II tractor lasts for two years. Switch points are the solution of the quadratic equation:

(a_I d_II - a_II d_I) R² + [a_I d_II - (d_II - d_I)] R - (d_II - d_I) = 0

where R = r + 1.

Various fluke switch points can be found for this special case. A fluke switch point exists at a rate of profits of -100 percent along the linear function:

c_II = [b_I c_I/(a_I b_II)] a_II

A fluke switch point exists on the wage axis when:

c_II = [(a_II)² b_I c_I - 2 a_II b_I c_I]/[2 a_I (a_I - 1) b_II]

A fluke switch point exists on the axis for the rate of profits when:

a_II = a_I (a_I + 1)

The fluke case in which the two wage curves are tangent at the switch point exists when the discriminant of the above quadratic equation is zero. That is, a reswitching pattern exists only if c_II is a solution of the following quadratic equation:

A (c_II)² + B c_II + C = 0

where:

A = (a_I)² (b_II)² (a_I + 1)²

B = -2 a_I a_II (1 + a_I + 2 a_II) b_I b_II c_I

C = (a_II)² (1 + 4 a_II) (b_I)² (c_I)²

I drew Figure 1 by numeric methods. The partitions agree with the above closed form expressions, which I later dervied.