Some Others Long Ago On Fluke Switch Points

I have been using fluke switch points to partition parameter spaces into regions. In each region, the analysis of the choice of technique does not qualitatively vary.

Fluke switch points have been discussed in the literature on the analysis of the choice of technique. Mostly, these mentions dismissal fluke cases, on the correct grounds that they only occur at an accidental point in the parameter space.

Here is a statement from one contribution to the famous QJE symposium:

"Cases with multiple roots or cases in which the curves cross only at end points … can be classified as irrelevant since the F[actor] P[rice] F[rontier] (envelope) is unchanged by their exclusion." -- Bruno, Michael, Edwin Burmeister, and Eytan Sheshinski. 1966. The nature and implications of the reswitching of techniques. Quarterly Journal of Economics, 80 (4): 534.

And another statement from the same contribution:

"'Adjacent' techniques on two sides of a switching point will usually differ from each other only with respect to one activity." -- Bruno, Michael, Edwin Burmeister, and Eytan Sheshinski. 1966. The nature and implications of the reswitching of techniques. Quarterly Journal of Economics, 80 (4): 542.

Here is from Garegnani's contribution:

"The possibility that, at r* and r**, the two curves touch without intersecting is excluded…" -- Garegnani, Pierangelo. 1966. Switching of techniques. Quarterly Journal of Economics, 80 (4): 567

I suppose I can look for more.

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.

Variations In An Analysis Of Intensive Rent With One Type Of Land (Part 2/2)

5.0 Fluke Cases

This post is a continuation of this one. This is a numeric example of intensive rent. Here I present five fluke cases before depicting how the analysis of the choice of technique varies with the full range of relative markups in agriculture.

5.1 Switch Point at Maximum Scale Factor for Epsilon

In the first fluke case, the wage curves for Alpha and Delta intersect at the maximum scale factor for the rate of profits for Delta (Figure 7). Figure 8 displays the graphs of the rent curves. At any larger scale factor, rent in Delta would be negative. This fluke case is associated with a qualitative change in the range of the scale factor for the rate of profits in which no cost- minimizing technique exists. The wage frontier consists of the wage curves for the Delta and Epsilon techniques up to the switch point between them. The wage frontier ends there. No technique is cost-minimizing for a scale factor between this switch point and the maximum scale factor for the rate of profits for Alpha.

Figure 7: Wage Curves for First Fluke Case

Figure 8: Rent Curves for First Fluke Case

This fluke case is associated with the disappearance of a range of the scale factor, for smaller relative markups in agriculture, in which only Alpha and Delta have positive scale factors for the rate of profits, and Delta has a positive rent. For a larger relative markup, a range of the scale factor appears in which only Alpha and Epsilon have positive scale factors, and Epsilon has a positive rent. A cost-minimizing technique exists in neither range.

5.2 Alpha vs. Epsilon Switch Point at Zero Wage

Another fluke case exists when the wage curves for Alpha and Epsilon intersect at a wage of zero. Figure 9 shows the wage curves, and Figure 10 shows the rent curves. In the last range for the scale factor, only Alpha can be under consideration for the cost-minimizing technique. For a smaller relative markup in agriculture, Epsilon is not eligible in this range because it would have a negative scale factor. For a larger relative markup, Epsilon is not eligible because it would have a negative rent.

Figure 9: Wage Curves for Second Fluke Case

Figure 10: Rent Curves for Second Fluke Case

These two fluke cases change some characteristics of the range of the scale factor of the rate of profits in which no cost- minimizing technique exists.

5.3 Switch Point for Three Techniques

For the next fluke case, all three wage curves, for Alpha, Delta, and Epsilon, intersect at a single switch point. Figures 11 and 12 show the wage and rent curves, respectively. This fluke case is associated with the disappearance of a range of the scale factor for the rate of profits in no technique is cost-minimizing even though both Alpha and Epsilon have a positive scale factor, and Epsilon has positive rent. It is also associated with the appearance of a range of the scale factor in which both Alpa and Delta are cost-minimizing.

Figure 11: Wage Curves for Third Fluke Case

Figure 12: Rent Curves for Third Fluke Case

5.4 Switch Point at Minimum Scale Factor for Delta

In the penultimate fluke case, Alpha and Epsilon have a switch point at the minimum scale factor for the rate of profits (Figures 13 and 14). This fluke case is associated with the disappearance of the range of the rate of profits at which both Epsilon and Delta are cost-minimizing.

Figure 13: Wage Curves for Fourth Fluke Case

Figure 14: Rent Curves for Fourth Fluke Case

5.5 Alpha vs. Delta Switch Point at Zero Wage

In the last fluke case (Fitures 15 and 16), Delta is only cost-minimizing at the switch point with Alpha. For a smaller scale factor, Delta does not have a non-negative rate of profits. For a larger scale factor, rent under Delta would be negative. This last fluke case is associated with the disappearance of a range of the scale factor for the rate of profits in which both Alpha and Delta are cost-minimizing.

Figure 15: Wage Curves for Fifth Fluke Case

Figure 16: Rent Curves for Fifth Fluke Case

6.0 All Markups in Agriculture

The above has briefly justified the vertical partitions in Figure 17, which shows the variation in the analysis of the cost-minimizing technique with perturbations of the markup up in agriculture. Table 3 shows how the analysis of the choice of technique varies among the numbered regions. If wants to look at these results in some detail, one can relate the variation in the analysis of the choice of technique to the fluke cases. This example demonstrates that my visualization techniques and perturbation analysis can be applied to an example where the cost-mninimizing technique is not found from a frontier of wage curves. The non-uniqueness and non-existence of a cost-minimizing technique arises in D'Agata's original example.

Figure 17: Variation of the Technique with the Markup in Agriculture

Table 3: The Cost-Minimizing Technique in Selected Regions in Parameter Space

Region	Range for Scale Factor	Cost-Minimizing Techniques
1	0 ≤ r ≤ Rδ	Epsilon
	Rδ ≤ r ≤ r*	Delta and Epsilon
	r* ≤ r ≤ Rε	None. Wage for Alpha, Delta, Epsilon positive. Rent for Delta and Epsilon positive.
	Rε ≤ r ≤ R*,δ	None. Wage for Alpha, Delta, positive. Rent for Delta positive.
	R*,δ ≤ r < Rα	None. Wage for Alpha positive.
2	0 ≤ r ≤ Rδ	Epsilon
	Rδ ≤ r ≤ r*	Delta and Epsilon
	r* ≤ r ≤ R*,δ	None. Wage for Alpha, Delta, Epsilon positive. Rent for Delta and Epsilon positive.
	R*,δ ≤ r ≤ Rε	None. Wage for Alpha, Epsilon positive. Rent for Epsilon positive.
	Rε ≤ r < Rα	None. Wage for Alpha positive.
3	0 ≤ r ≤ Rδ	Epsilon
	Rδ ≤ r ≤ r*	Delta and Epsilon
	r* ≤ r ≤ R*,δ	None. Wage for Alpha, Delta, Epsilon positive. Rent for Delta and Epsilon positive.
	R*,δ ≤ r < R*,ε	None. Wage for Alpha and Epsilon positive. Rent for Epsilon positive.
	R*,ε ≤ r < Rα	None. Wage for Alpha positive.
4	0 ≤ r ≤ Rδ	Epsilon
	Rδ ≤ r ≤ R*,ε	Delta and Epsilon
	R*,ε ≤ r ≤ R*,δ	Alpha and Delta
	R*,δ ≤ r < Rα	None. Wage for Alpha positive.
5	0 ≤ r ≤ R*,ε	Epsilon
	R*,ε ≤ r ≤ Rδ	Alpha
	Rδ ≤ r ≤ R*,δ	Alpha and Delta
	R*,δ ≤ r < Rα	None. Wage for Alpha positive.
6	0 ≤ r ≤ R*,ε	Epsilon
	R*,ε ≤ r ≤ Rα	Alpha

7.0 Conclusion

The analysis of the choice of technique with rent, including intensive rent, is more complicated than such analysis in a model with only circulating capital:

"The complexity of the outcomes with the potential existence of conflict or concordance among the three major economic categories (earners of wages, profits, and rents) profoundly modifies the traditional analysis of profits and wages." -- Alberto Quadrio Curzio and Fausta Pellizzari (2010).

The quantity and price systems are interconnected. Assumptions on the level of net output are required to determine which techniques are feasible. Introducing relative market power among industries further complicates the analysis of the choice of technique.

Variations In An Analysis Of Intensive Rent With One Type Of Land (Part 1/2)

Figure 1: Variation of the Technique with the Markup in Agriculture

1.0 Introduction

This post is the start of a recreation of a previous post, with a requirement that relative markups lie on a simplex.

These two posts are intended to explain Figure 1, above, which presents a summary of the results of the analysis of the choice of technique, given any level of the relative markup in agriculture, as compared with the relative markups in the non-agricultural industries. My presentation is long enough that I break in down into a couple of posts.

The numerical example illustrates that the interest of landlords are affected by persistent barriers of entry in industry and agriculture, as well as class struggle between workers and capitalists. Other classes care about attempts among capitalists to establish non-competitive market structures, although not in any transparent way.

The example also illustrates the possibility of the existence of multiple cost-minimizing techniques away from switch points and of the non-existence of a cost-minimizing technique. In the first case, a finite number of long-period positions are consistent with a given rate of profits, so to speak. The results have a certain indeterminancy, in this sense. The non-existence of a cost-minimizing technique is compatible with the existence of feasible techniques that yield a positive wage and positive prices of production.

Multiple cost-minimizing techniques and the non-existence of a cost-minimizing technique are possibilities in the theory of joint production. They cannot arise with circulating capital alone, pure fixed capital models, and certain models of extensive rent.

This model is not structured so as to be able to yield variation in the order of rentability with distribution or of the reswitching of the order of rentability.

2.0 Technology, Endowments, and Requirements for Use

I might as well repeat the data. Table 1 shows the coefficients of production. Only one type of land exists, and three processes are known for producing corn on it. Following D'Agata, assume that one hundred acres of land are available and that net output consists of 90 tons iron, 60 tons steel, and 19 bushels corn. The net output is also the numeraire.

Table 1: The Coefficients of Production

Input	Industry
	Iron	Steel	Corn
	I	II	III	IV	V
Labor	1	1	1	11/5	1
Land	0	0	1	1	1
Iron	0	0	1/10	1/10	1/10
Steel	0	0	2/5	1/10	1/10
Corn	1/10	3/5	1/10	3/10	2/5

All three commodities must be produced for any composition of net output. Table 2 lists the available techniques. Only Alpha, Delta, and Epsilon are feasible for these requirements for use. Not all land is farmed and only one corn-producing process is operated under Alpha. Two corn-producing processes are operated together under Delta and Epsilon.

Table 2: Techniques

Technique	Processes
Alpha	I, II, III
Beta	I, II, IV
Gamma	I, II, V
Delta	I, II, III, IV
Epsilon	I, II, III, V
Zeta	I, II, IV, V

In the non-competitive case, the relative markups in different industries are taken as given. Let the rates of profits be in proportions of s₁, s₂, and s₃, respectively.

3.0 Prices of Production

Prices of production can be defined for each technique. In what is my usual notation, prices of production must satisfy the following system of equations for Delta:

(p₁ a_1,1 + p₂ a_2,1 + p₃ a_3,1)(1 + s₁ r) + w a_0,1 = p₁

(p₁ a_1,2 + p₂ a_2,2 + p₃ a_3,2)(1 + s₂ r) + w a_0,2 = p₃

(p₁ a_1,3 + p₂ a_2,3 + p₃ a_3,3)(1 + s₃ r) + phi c_1,3 + w a_0,3 = p₃

(p₁ a_1,4 + p₂ a_2,4 + p₃ a_3,4)(1 + s₃ r) + phi c_1,4 + w a_0,4 = p₃

The wage is w. I call r the scale factor for the rate of profits. I assume that the same rate of profits, s₃ r, is obtained in both corn-producing processes operated under Delta. The requirements for use yield an equation for the numeraire:

p₁ d₁ + p₂ d₂ + p₃ d₃ = 1

Lately, I have been imposing the condition that relative markups lie on a simplex:

s₁ + s₂ + s₃ = 1

For the analyses in these posts, I assume that relative markups are the same in both industrial sectors:

s₁ = s₂

The above system of equations are such that the three prices of produced commodities, the wage, and rent per acre can all be expressed as a function of the scale factor for the rate of profits, given the technique. Which technique is cost-minimizing at any given scale factor?

4.0 Competitive Markets

To begin, consider the special case of competitive markets. No barriers to entry or exit exist, or any other mechanism that keeps the rate of profits persistently unequal among industries. In the notation in this example:

s₁ = s₂ = s₃ = 1/3

Figure 2 shows the resulting wage curves for the feasible techniques, and Figure 3 shows the corresponding rent curves. For a small enough scale factor for the rate of profits, the wage for Delta is not non-negative. In this range, Epsilon is cost-minimizing, and land obtains a rent. In the range of the scale factor in which the wage is positive for Delta, up to the switch point between Delta and Epsilon are both cost-minimizing. Landlords would rather have the capitalists adopt Delta, but the model is silent on which cost-minimizing technique will be adopted. For any larger scale factor for the rate of profits, no cost-minimizing technique exists. One could break down this range into three subranges, depending on whether both Delta and Epsilon pay positive rents, only Epsilon pays a positive rent, or neither do. By the way, the competitive case is in region 2 in the figure at the top of this post.

Figure 2: Wage Curves with Competitive Markets

Figure 3: Rent Curves with Competitive Markets

It remains to justify the claims above about which feasible technique is cost-minimizing, given the scale factor for the rate of profits. Suppose prices of production for Alpha prevail. One can calculate, for each of the five processes, the difference between the price of the output and the costs of the inuts (Figure 4). Inputs of iron, steel, and corn are costed up at the going scaled rate of profits in each process. Wages are paid out the surplus at the end of the period. The difference, in each process, is known as supernormal or extra profits.

Figure 4: Extra Profits with Alpha Prices with Competitive Markets

Figure 4 shows that Alpha is never cost-minimizing. Whatever the scale factor, extra profits can always be obtained by growing corn with the fifth process. If the rate of profits is high eneough, extra profits can also be obtained by growing corn with the fourth process. Some capitalists would soon adopt another process to produce corn if Alpha were in operation.

How about if the Delta technique were in operation? Both the third and fourth process would be operated. All land would be farmed, and land would obtain a rent. Figure 5 shows the extra profits in operating the last process under these conditions. (By the way, I always try to draw graphs like these to check the solutions of the price equations.) You can see that for a scale factor for the rate of profits greater than at the switch point with Epsilon, Delta is not cost-minimizing. For a lower rate of profits, Delta is cost-minimizing in the range in which the wage is non-negative.

Figure 5: Extra Profits with Delta Prices with Competitive Markets

Finally, consider prices of production for Epsilon. Figure 6 plots supernormal profits for each process with these prices. Epsilon is cost-minimizing up to the switch point with Delta. Beyond this, price signals indicate that the forth process should be operated. Neither Delta nor Epsilon, much less Alpha, are cost-minimizing in this range.

Figure 6: Extra Profits with Epsilon Prices with Competitive Markets

The above has shown which feasible techniques are cost-minimizing, if any, for the full range of the rate of profits in the competitive case. This analysis essentially replicates D'Agata's example. The next part considers non-competitive markets, including fluke cases in which there are qualitative changes in the analysis of the choice of technique.

Elsewhere

• Ramon Llul was a medieval scholar who invented social choice theory (specifically, the Borda count), as seen in a manuscript discovered in 2001(?). (Other discoveries.)
• A review of Steve Paxton's Unlearning Marx: Why Soviet Failure was a Triumph for Marx. Maybe I want to read this.
• Pierangelo Garegnani's Capital Theory, the Surplus Approach, and Effective Demand.

Welcome

I study economics as a hobby. My interests lie in Post Keynesianism, (Old) Institutionalism, and related paradigms. These seem to me to be approaches for understanding actually existing economies.

The emphasis on this blog, however, is mainly critical of neoclassical and mainstream economics. I have been alternating numerical counter-examples with less mathematical posts. In any case, I have been documenting demonstrations of errors in mainstream economics. My chief inspiration here is the Cambridge-Italian economist Piero Sraffa.

In general, this blog is abstract, and I think I steer clear of commenting on practical politics of the day.

I've also started posting recipes for my own purposes. When I just follow a recipe in a cookbook, I'll only post a reminder that I like the recipe.

Comments Policy: I'm quite lax on enforcing any comments policy. I prefer those who post as anonymous (that is, without logging in) to sign their posts at least with a pseudonym. This will make conversations easier to conduct.

Variations In Switch Points With Markups In The 'Corn' Industry

Figure 1: Variation of Switch Points with the Markup in the Corn Industry

1.0 Introduction

I have been re-creating some of my past analyses. The graphs in this post look a bit different because I impose a requirement that the relative markups sum to unity.

2.0 Technology

Consider an economy which produces three commodities, iron, steel, and corn, with the technology specified in Table 1. Two processes are available for producing each commodity. The coefficients of production in a column specify the person-years of labor, tons of iron, tons of steel, and bushels of corn required to produce a unit of output of the given industry.

Table 1: The Technology

Input	Iron Industry		Steel Industry		Corn Industry
	a	b	c	d	e	f
Labor	1/3	1/10	5/2	7/20	1	3/2
Iron	1/6	2/5	1/200	1/100	1	0
Steel	1/200	1/400	1/4	3/10	0	1/4
Corn	1/300	1/300	1/300	0	0	0

Eight techniques (Table 2) are defined for this technology. Each technique is defined by the operation of one process in each of the three industries. All three commodities are Sraffian basics in all techniques. That is, each commodity is a direct or indirect input in the production of all commodities. For example, iron is used directly as an input in the first corn-producing process, and steel is used indirectly in producing corn with this process since steel is an input in either iron-producing process

Table 2: Techniques

Technique	Processes
Alpha	a, c, e
Beta	a, c, f
Gamma	a, d, e
Delta	a, d, f
Epsilon	b, c, e
Zeta	b, c, f
Eta	b, d, e
Theta	b, d, f

3.0 Prices of Production

Prices of production are defined here for given ratios of markups among industries. The ratios of rates of profits among industries are assumed stable, but rates of profits are not necessarily uniform. Lack of uniformity in rates of profits can result from variations in evaluations of profits among industries due to idiosyncratic properties of investment; from barriers to entry arising from, for example, secrets in manufacture; and from legal monopolies (D’Agata 2018). Let s₁ r, s₃ r, and s₃ r be the rate of profits in the iron, steel, and corn industries respectively. I call r the scale factor for the rate of profits. The usual system of equations, with labor advanced, must be satisfied for prices of production for a given technique.

As a matter of scaling, suppose the markups lie on a simplex:

s₁ + s₂ + s₃ = 1

Suppose that a bushel of corn is the numeraire. In drawing various graphs, I consider only variations in the markup in the corn industry, with markups in producing iron and steel assumed identical:

s₁ = s₂

The solution to this system, for each technique, has a single degree of freedom, which can be expressed with the wage as a function of the scale factor for the rate of profits

4.0 The Choice of Technique with Competitive Markets

Figure 1 graphs the wage curves for four techniques, given competitive markets. The same relative markups are obtained in all industries. The cost-minimizing technique at a given wage maximizes the scale factor for the rate of profits. The cost-minimizing technique at a given scale factor maximizes the wage. The outer frontier of all wage curves shows the variation of the cost-minimizing technique with distribution. Wage curves are graphed in Figure 1 only for the techniques on the outer frontier. This type of figure, usually for competitive markets, is the most well-known graph in post-Sraffian price theory

Figure 2: Capital-Reversing with Competitive Markets

Around the so-called perverse switch point, the firms in the corn industry switch from the second corn-producing process to the first at a lower wage. That is, they adopt a process that requires less labor to be hired per bushel of corn produced gross. This is known as the reverse substitution of labor (Han and Schefold 2006). For the economy as a whole, the technique adopted at a lower wage requires less labor per unit of net output. This is a consequence of capital-reversing as manifested in a comparison of stationary states (Harris 1973).

5.0 Fluke Cases

Five fluke cases can be found by perturbing the relative markup in the corn industry (Table 3). Figure 3 depicts the wage frontier for the first fluke case. This markup occurs when reswitching is just emerging.

Table 3: Fluke Switch Points

Markup for Corn	Fluke Case
s3 ≈ 0.211996	Reswitching pattern for Gamma vs. Delta.
s3 ≈ 0.249246	Four technique pattern for Gamma, Delta, Eta, and Theta.
s3 ≈ 0.8232415	Alpha vs Beta switch point at wage of zero.
s3 ≈ 0.8696757	Four technique pattern for Alpha, Beta, Gamma, and Delta.
s3 ≈ 0.9307414	Beta vs Delta pattern over r axis

Figure 3: Wage Curves for Gamma and Delta Tangent at Switch Point

6.0 The Choice of Technique with the Full Range of the Markup in the Corn Industry

Figure 1, at the top of this post, is my new type of diagram illustrated for depicting the analysis of the choice of technique. The abscissa is the markup in the corn industry, with given markups of unity in the iron and steel industry. The maximum wage and the wage at switch points along the frontier are plotted. The number and sequence of switch points along the wage frontier are invariant in each numbered region. Fluke switch points partition the numbered regions. Figure 4 enlarges Figure 1 on the right for low wages

Figure 4: Variation of Switch Points with the Markup (Detail)

The qualitative properties of the wage frontier are invariant in each numbered region in Figures 1 and 4. Table 4 describes each numbered region. The cost-minimizing technique along the wage frontier is listed, from a wage of zero to the maximum wage. Some salient properties of switch points and the cost-minimizing technique are summarized in Table 5. Figure 2 depicts the wage frontier for a markup in the corn-industry in region 3, while Figure 3 depicts the wage frontier on the boundary between regions 1 and 2.

Table 4: Variations in the Cost-Minimizing Technique

Region	Range	Technique
1	0 ≤ w ≤ w1	Alpha
	w1 ≤ w ≤ w2	Gamma
	w2 ≤ w ≤ wmax,η	Eta
2	0 ≤ w ≤ w1	Alpha
	w1 ≤ w ≤ w2	Gamma
	w2 ≤ w ≤ w3	Delta
	w3 ≤ w ≤ w4	Gamma
	w4 ≤ w ≤ wmax,η	Eta
3	0 ≤ w ≤ w1	Alpha
	w1 ≤ w ≤ w2	Gamma
	w2 ≤ w ≤ w3	Delta
	w3 ≤ w ≤ w4	Theta
	w4 ≤ w ≤ wmax,η	Eta
4	0 ≤ w ≤ w1	Beta
	w1 ≤ w ≤ w2	Alpha
	w2 ≤ w ≤ w3	Gamma
	w3 ≤ w ≤ w4	Delta
	w4 ≤ w ≤ w5	Theta
	w5 ≤ w ≤ wmax,η	Eta
5	0 ≤ w ≤ w1	Beta
	w1 ≤ w ≤ w2	Delta
	w2 ≤ w ≤ w3	Theta
	w3 ≤ w ≤ wmax,η	Eta
6	0 ≤ w ≤ w1	Delta
	w1 ≤ w ≤ w2	Theta
	w2 ≤ w ≤ wmax,η	Eta

Table 5: Notes on Regions

Region	Summary
1	No reswitching, no capital-reversing, no reverse substitution of labor, no process recurrence.
2	Reswitching of techniques between Gamma and Delta. Capital-reversing and the reverse substitution of labor at the switch point between Gamma and Delta at the lower wage. Process recurrence of the first process in the corn industry.
3	No reswitching. Capital-reversing and the reverse substitution of labor at the switch point between Gamma and Delta. Process recurrence of the first process in the corn industry.
4	No reswitching. Capital-reversing and the reverse substitution of labor at the switch point between Gamma and Delta. Process recurrence of both processes in the corn industry.
5	No reswitching, no capital-reversing, no reverse substitution of labor, no process recurrence.
6	No reswitching, no capital-reversing, no reverse substitution of labor, no process recurrence.

This example allows for a graphical display showing that reswitching arises with an increased markup in corn-production, starting from a markup much less than in other industries. The ‘perverse’ switch point between Gamma and Delta remains on the wage frontier after the other switch point between these techniques falls off the frontier at a higher markup. Eventually, the ‘perverse’ switch point is no longer on the frontier when corn-production has a much higher markup than other industries.

7.0 Conclusion

The properties of the wage frontier might be thought to have some impact on the struggle between capitalists and workers. These properties can be altered both by technical change and by variations in relative market power among capitalists.