Local Perturbations Of A Fluke Switch Point For Intensive Rent

Figure 1: A Parameter Space

1.0 Introduction

This is a re-creation and elaboration of a previous post.

The analysis of the choice of technique, in models of circulating and fixed capital, can be based on the construction of a wage-rate of profits frontier. Given a technology in which requirements for use can be satisfied, prices of production for a feasible technique, including the wage, are uniquely determined by the given rate of profits. If the rate of profits is in a range where such prices are non-negative for at least one technique, one of the techniques is uniquely cost-minimizing, except at switch points. These properties do not necessarily hold in models of general joint production. An examination of local perturbations in an example of intensive rent illustrates surprising possibilities.

2.0 Technology

Table 1 presents coefficients of production, a perturbation of an example from D'Agata (1983). Only one type of land exists, and three processes are known for producing corn on it. The scarcity of land is shown by the possibility of two corn-producing processes being operated side-by-side in the cost-minimizing technique.

Table 1: Coefficients of Production

Inputs	Industry
	Iron	Steel	Corn
	I	II	III	IV	V
Labor	1	1	1	57/20	21/20
Land	0	0	1	13/10	21/20
Iron	0	0	1/10	a1,4	a1,5
Steel	0	0	2/5	13/100	21/200
Corn	1/10	3/5	1/10	2/5	21/50

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. 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 the parameter ranges considered. 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 of Production

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

3.0 A Fluke Switch Point

Figure 2 shows the wage and rent curves for feasible techniques at a selected parametrization. I take the wage curve for a technique to be defined only for non-negative rates of profits at which the wage, rent per acre, and the prices of produced commodities are non-negative. The wage is negative for Delta for rates of profits below that at the switch point, and rent is negative for rates of profits greater than that at the switch point. Rent is negative for Epsilon for rates of profits less than at the switch point, and the wage is negative for greater rates of profits. Thus, the switch point is the only point on the wage curves for Delta and Epsilon. The switch point is a fluke in at least two ways. It is a switch point for three techniques, not two. And it is on the axis for the rate of profits.

Figure 2: Wage and Rent Curves for a Fluke Case

4.0 Local Perturbations within a Parameter Sapce

Figure 1, at the top of this post, shows a partition of the parameter space around this fluke case. An intersection of three wage curves over the axis for the rate of profits is a combination of three pairs of wage curves intersecting over the axis for the rate of profits. These three fluke cases are the partitions between regions 1 and 2, regions 2 and 3, regions 6 and 7, regions 7 and 8, and regions 8 and 1. The partition between regions 3 and 4 is associated with the fluke case of three wage curves intersecting at a non-negative rate of profits. The partitions between regions 4 and 5 and between regions 5 and 6 illustrate a fluke switch point specific to models of rent.

Regions 2 through 8 illustrate the possible non-uniqueness and non-existence of a cost-minimizing technique. For concreteness, consider the point in region 5 with the wage curves and variation in rent per acre illustrated in Figure 3. For rates of profits up to the first switch point, Alpha is cost-minimizing. Epsilon is cost-minimizing between the switch points, and Delta is also cost-minimizing for high rates of profits in this range. Beyond the second switch point, no technique is cost-minimizing. Whether or not land is scarce depends on the distribution of income.

4.1 The Choice of Technique in Region 5

Figure 3: Wage and Rent Curves in Region 5

How can one determine which techniques are cost-minimizing for a given rate of profits? Given the technique and the rate of profits, the costs of the capital goods, the rent on land, and wages can be summed for a unit level for each process. Iron, steel, and corn inputs incur the going rate of profits in this sum. The difference between the revenues and this sum is the extra profits obtained in operating a process. By definition, no process comprising the technique yields extra profits. The technique is cost-minimizing if extra profits cannot be obtained by operating any other process

For the parameters illustrated in Figure 3, extra profits are obtained by operating process IV or V at Alpha prices for a rate of profits greater than that at the first switch point. Alpha is only cost minimizing at a lower rate of profits. Figure 4 depicts the extra profits available from the last two corn-producing process at Delta and Epsilon prices. The range of rates of profits in which each technique is cost-minimizing is indicated, and these ranges overlap. For rates of profits immediately greater than the rate of profits at the second switch point, prices of production indicate that Epsilon should be adopted when prices of production for Delta prevail, and that Delta should be adopted when prices of production for Epsilon prevail. This circuit is a manifestation of the non-existence of a cost-minimizing technique.

Figure 4: Extra Profits for Delta and Epsilon in Region 5

4.2 Fluke Cases Bordering Region 5

Presenting two fluke switch points might assist in understanding how the analysis of the choice of technique varies in the part of the parameter space examined here. Figure 5 shows the wage and rent curves for a fluke switch point for parameters on the partition between regions 4 and 5. This fluke switch point is associated with the disapperance of the range of the rate of profits, in region 4, where both Alpha and Epsilon are cost-minimizing. It is associated with the emergence, in region 5, of a range of the rate of profits where only Epsilon is cost-minimizing.

Figure 5: A Fluke Switch Point on the Upper Boundary of Region 5

Figure 6, on the other hand, shows a fluke switch point for parameters on the boundary of regions 5 and 6. This switch point is associated with the disappearance of a range of the rate of profits where only Alpha and Epsilon have defined wage and rent curves and neither technique is cost-minimizing. And it is associated with the appearance of a range of the rate of profits where only Alpha and Delta have defined wage and rent curves and neither technique is cost-minimizing. The fluke switch points in Figure 5 and Figure 6 can only arise in models of joint production, including models of rent.

Figure 6: A Fluke Switch Point on the Lower Boundary of Region 5

4.3 Overview of Regions as a Whole

Qualitative properties of the analysis of the choice of technique do not vary within each numbered region. Table 3 describes the variation in the cost-minimizing technique with the rate of profits in each numbered region in Figure 1. In region 1, Alpha is cost-minimizing for all feasible rates of profits. Land is not scarce, and obtains no rent. For a high enough rate of profits in region 2, Alpha and Delta are both non-uniquely cost-minimizing. The wage curve for Delta slopes up and rent per acre decreases with the rate of profits when Delta is operated. The switch point for Alpha and Delta is at a positive wage. For any rate of profits greater than the rate of profits at the switch point, no cost-minimizing technique exists. In region 3, a switch point between Alpha and Epsilon occurs at a rate of profits higher than the maximum rate of profits for Delta. Epsilon is never cost-minimizing.

Table 3: Ranges of the Rate of Profits by Region

Region	Range	Technique	Comment
1	0 ≤ r ≤ rα,max	Alpha	Delta and Epsilon have negative wage or rent throughout.
2	0 ≤ r ≤ rδ,min	Alpha	Alpha has a positive wage.
	rδ,min ≤ r ≤ rδ,max	Alpha & Delta	Alpha has a positive wage; Delta has a positive wage and rent.
	rδ,max ≤ r ≤ rα,max	None	Alpha has a positive wage.
3	0 ≤ r ≤ rδ,min	Alpha	Alpha has a positive wage.
	rδ,min ≤ r ≤ rδ,max	Alpha & Delta	Alpha has a positive wage; Delta has a positive wage and rent.
	rδ,max ≤ r ≤ rε,min	None	Alpha has a positive wage.
	rε,min ≤ r ≤ rε,max	None	Alpha has a positive wage; Epsilon has a positive wage and rent.
	rε,max ≤ r ≤ rα,max	None	Alpha has a positive wage.
4	0 ≤ r ≤ rδ,min	Alpha	Alpha has a positive wage.
	rδ,min ≤ r ≤ rε,min	Alpha & Delta	Alpha has a positive wage; Delta has a positive wage and rent.
	rε,min ≤ r ≤ r1	Delta & Epsilon	Alpha has a positive wage; Delta and Epsilon have a positive wage and rent.
	r1 ≤ r ≤ rδ,max	None	Alpha has a positive wage; Delta and Epsilon have a positive wage and rent.
	rδ,max ≤ r ≤ rε,max	None	Alpha has a positive wage; Epsilon has a positive wage and rent.
	rε,max ≤ r ≤ rα,max	None	Alpha has a positive wage.
5	0 ≤ r ≤ rε,min	Alpha	Alpha has a positive wage.
	rε,min ≤ r ≤ rδ,min	Epsilon	Alpha has a positive wage; Epsilon has a positive wage and rent.
	rδ,min ≤ r ≤ r1	Delta & Epsilon	Alpha has a positive wage; Delta and Epsilon have a positive wage and rent.
	r1 ≤ r ≤ rδ,max	None	Alpha has a positive wage; Delta and Epsilon have a positive wage and rent.
	rδ,max ≤ r ≤ rε,max	None	Alpha has a positive wage; Epsilon has a positive wage and rent.
	rε,max ≤ r ≤ rα,max	None	Alpha has a positive wage.
6	0 ≤ r ≤ rε,min	Alpha	Alpha has a positive wage.
	rε,min ≤ r ≤ rδ,min	Epsilon	Alpha has a positive wage; Epsilon has a positive wage and rent.
	rδ,min ≤ r ≤ r1	Delta & Epsilon	Alpha has a positive wage; Delta and Epsilon have a positive wage and rent.
	r1 ≤ r ≤ rε,max	None	Alpha has a positive wage; Delta and Epsilon have a positive wage and rent.
	rε,max ≤ r ≤ rδ,max	None	Alpha has a positive wage; Delta has a positive wage and rent.
	rδ,max ≤ r ≤ rα,max	None	Alpha has a positive wage.
7	0 ≤ r ≤ rε,min	Alpha	Alpha has a positive wage.
	rε,min ≤ r ≤ rε,max	Epsilon	Alpha has a positive wage; Epsilon has a positive wage and rent.
	rε,max ≤ r ≤ rδ,min	None	Alpha has a positive wage.
	rδ,min ≤ r ≤ rδ,max	None	Alpha has a positive wage; Delta has a positive wage and rent.
	rδ,max ≤ r ≤ rα,max	None	Alpha has a positive wage.
8	0 ≤ r ≤ rε,min	Alpha	Alpha has a positive wage.
	rε,min ≤ r ≤ rε,max	Epsilon	Alpha has a positive wage; Epsilon has a positive wage and rent.
	rε,max ≤ r ≤ rα,max	None	Alpha has a positive wage.

In region 4, a switch point exists on the wage frontier between Alpha and Epsilon, at a rate of profits greater than the minimum rate of profits for Delta. A range of the rate of profits remains at which Alpha and Delta are both non-uniquely cost-minimizing. Above the rate of profits at this switch point, the wage frontier resembles the wage frontier in Figure 3 at rates of profits greater than the minimum rate of profits for Delta. In region 5, the range of the rate of profits at which both Alpha and Delta are cost-minimizing has disappeared. In region 6, the range of the rate of profits has disappeared in which no technique is cost minimizing, but Epsilon has a positive rate of profits and rent and Delta does not.

In region 7, Delta is no longer cost-minimizing at any feasible rate of profits. Alpha is cost-minimizing at a low rate of profits, and Epsilon is uniquely cost-minimizing at any feasible rate of profits greater than the rate of profits at the switch point between Alpha and Epsilon. In region 8, Delta is not only no longer ever cost-minimizing, but Delta never has both a positive rate of profits and rent.

Suppose one is not interested in qualitative variations in ranges of the rate of profits in which no cost-minimizing technique exists. In one range for some regions, Alpha has a positive rate of profits, and Delta and Epsilon each have positive rates of profits and a positive rent. Yet when prices for Delta prevail, extra profits can be obtained by operating processes in Epsilon. And when prices for Epsilon prevail, extra profits come from adopting Delta. In another range of rates of profits, neither Delta nor Epsilon obtain both non-negative rates of profits and rents. Yet Alpha is not cost-minimizing. Ignoring these variations, regions 2 and 3 can be combined. Regions 5 and 6 can be combined. Likewise, regions 7 and 8 can be combined

5.0 Conclusion

Whether or not land obtains a rent can depend on the distribution of income. For a low-enough rate of profits in regions 2 through 8, the first three processes are operated. Iron, steel, and corn are each produced with one process, and land obtains no rent. For a higher rate of profits, the Delta or Epsilon technique can be cost-minimizing. Corn is produced by two processes, and scarce land obtains a rent. Even if the requirements for use can feasibly be satisfied with some land not farmed, the cost-minimizing technique may be such that two processes are operated side-by-side on land, with no land lying fallow. The example illustrates that an examination of fluke switch points can help in understanding qualitative variations in the analysis of the choice of technique, even in a case where certain properties of models of circulating capital do not hold.

References

• D'Agata, Antonio. 1983. The existence and unicity of cost-minimizing systems in intensive rent theory. Metroeconomica, 35: 147-158.
• Kurz, Heinz and Neri Salvadori. 1995. Theory of Production: A Long-Period Analysis. Cambridge: Cambridge University Press.

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.

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