Three Examples For The Cambridge Capital Controversy

Figure 1: A Parameter Space

1.0 Introduction

I have been reconstructing some of my examples. The first example in this post is from here. I am thinking of writing a draft article, as mentioned here. While I am at it, I thought I would also work through the examples in Garegnani (1966) and Bruno, Burmeister & Sheshinski (1966), both from the symposium in the Quarterly Journal of Economics of that year.

2.0 The Emergence of the Reverse Substitution of Labor

This section presents an example with circulating capital alone. Table 1 presents the technology for an economy in which two commodities, iron and corn, are produced. Managers of firms know of one process for producing iron and two for producing corn. Each process is specified by coefficients of production, that is, the required physical inputs per unit output. The Alpha technique consists of the iron-producing process and the first corn-producing process. Similarly, the Beta technique consists of the iron-producing process and the second corn-producing process. At any time, managers of firms face a problem of the choice of technique

Table 1: Technology for the Reverse Substitution of Labor

Input	Industry
	Iron	Corn
		Alpha	Beta
Labor	a0,1=1	aα0,2=16/25	aβ0,2
Iron	a1,1=9/20	aα1,2=1/625	aβ1,2
Corn	a2,1=2	aα2,2=12/25	aβ2,2=27/400

Two parameters are not given numerical values in this specification of technology. The approach taken here is to examine a local perturbation of parameters in a two-dimensional slice of the higher dimensional parameter space defined by the coefficients of production in particular numeric examples. With wages paid out of the surplus product at the end of the period of production, the wage curves for the two techniques are depicted in Figure 2 for a particular parametrization of the coefficients of production. The Beta technique is cost-minimizing for any feasible distribution of income. If the wage is zero and the workers live on air, the Alpha technique is also cost-minimizing.

Figure 2: Wage Curves with Two Fluke Switch Point

A switch point is defined in this model of circulating capital to be an intersection of the wage curves. These switch points, for the particular parameter values illustrated in Figure 2, are fluke cases. Almost any variation in the model parameters destroys their interesting properties. A switch point exists at a rate of profits of -100 percent only along a knife edge in the parameter space (Figure 1). Likewise, a switch point exists on the axis for the rate of profits only along another knife edge. The illustrated example, with two fluke switch points, arises at a single point in the parameter space, where these two partitions intersect.

Figure 1 depicts a partition of the parameter space around the point with these two fluke switch points. Below the horizontal line, the switch point on the axis for the rate of profits has disappeared below the axis. The Beta technique is cost-minimizing for all feasible non-negative rates of profits. Above this locus, the Alpha technique is cost-minimizing for a low enough wage or a high enough feasible rate of profits.

In the northwest, the switch point at a negative rate of profits occurs at a rate of profits lower than 100 percent. Around the switch point at a positive rate of profits, a lower wage is associated with the adoption of the corn-producing process with a larger coefficient for labor. That is, at a higher wage, employment is lower per unit of gross output in the corn industry.

In the northeast of Figure 1, the switch point for a positive rate of profits exhibits the reverse substitution of labor. Around this switch point, a higher wage is associated with the adoption of a process producing the consumer good in which more labor is employed per unit of gross output. The other switch point exists for a rate of profits between -100 percent and zero. Steedman (2006) presents examples with this phenomenon in models with other structures

Qualitative changes in the wage frontier exist in the parameter space away from the part graphed in Figure 1. The analysis presented here is of local perturbations of the depicted fluke case.

2.0 Example from Garegnani (1966)

I think of Luigi Pasinetti as the first to show that David Levhari's non-(re)switching theorem is false. But the counter-example that he presented at the September 1965 Rome Congress of the Econometric Society did not quite meet all of the assumptions of Levhari's theorem.

Table 2 defines the coefficients of production for the counter-example from Pierangelo Garegnani's paper in the QJE symposium devoted to the topic. Figure 3 presents the wage curves for the example. Switch points are at 10 percent and 20 percent, appealingly reasonably small rates of profits. But the wage curves are visually hard to distinguish. The switch points are more apparent in the plot of extra profits at Alpha prices, in the right pane.

Table 2: Technology for a Reswitching Example

Input	Industry
	Iron	Corn
		Alpha	Beta
Labor	a0,1=89/10	aα0,2=9/50	aβ0,2=3/2
Iron	a1,1=0	aα1,2=1/2	aβ1,2=1/4
Corn	a2,1=379/423	aα2,2=1/10	aβ2,2=5/12

Figure 3: Wage Curves for a Reswitching Example

In some sense, it is unfair to criticize scholars of that time for not creating more apparent examples. The tools I have are much more advanced for seeing the effect of perturbing a coefficient. And, nevertheless, I still have some examples that are hard to see the 'perverse' results.

3.0 Example from Bruno, Burmeister & Sheshinski (1966)

The counter example from Michael Bruno, Edwin Burmeister, and Eytan Sheshinski's paper in the QJE symposium has more a visually striking wage frontier. Table 3 presents the coefficients of production. (I have reordered the industries.) Figure 4 plots the wage curves. The switch points are at approximately 46.58 percent and 166.88 percent or wages of approximately 0.8065 and 0.2595 bushels per person-year.

Table 3: Technology for Another Reswitching Example

Input	Industry
	Iron	Corn
		Alpha	Beta
Labor	a0,1=1	aα0,2=33/100	aβ0,2=1/100
Iron	a1,1=0	aα1,2=1/50	aβ1,2=71/100
Corn	a2,1=1/10	aα2,2=3/10	aβ2,2=0

Figure 4: Wage Curves for another Reswitching Example

Many like to quote Paul Samuelson declaration that:

"...the simple tale told by Jevons, Böhm-Bawerk, Wicksell, and other neoclassical writers - alleging that, as interest rate falls in consequence of abstention from present consumption in favor of future, technology must become in some sense more 'roundabout,' more 'mechanized,' and more 'productive' - cannot be universally valid." -- Paul A. Samuelson (1966).

Bruno, Burmeister & Sheshinski are just as clear:

"Numerical examples and the realization that switching points are roots of n-th degree polynomials (and therefore numerous) have convinced us that reswitching may well occur in a general capital model." - Bruno, Burmeister & Sheshinski (1966, p. 527)

Somehow, empirical work has not made it apparent all of these possible real roots, despite the exploration of economies with many industries. I like this quotation too:

"Although the latter sufficiency condition is again highly restrictive, it may be somewhat less restrictive than the former one: note the latter allows changes of single activities while the former does not. We might also observe that the latter condition seems to be the most natural extension of our previous two-sector nonswitching theorem... Let us again stress that, except for highly exceptional circumstances, techniques cannot be ranked in order of capital intensity. We thus conclude that reswitching is, at least theoretically; a perfectly acceptable case in the discrete capital model." - Bruno, Burmeister & Sheshinski (1966, p. 545)

I skimmed the sufficiency condition. I think technologies with different capital goods used in different techniques are ruled out. Likewise, processes in the same industry in which some capital goods are increased and others are decreased might also be ruled out. It is the general case that technology can be such that reswitching is possible.