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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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Another Example Of A Real Wicksell Effect Of Zero

Figure 1: A Reswitching Example with a Fluke Switch Point

1.0 Introduction

A switch point that occurs at a rate of profits of zero is a fluke. This post presents a two-commodity example with a choice between two techniques, in which restitching occurs, and one switch point is such a fluke. Total employment per unit of net output is unaffected by the choice of technique. Furthermore, the numeraire-value of capital per unit net output is also unaffected by the mix of techniques adopted at a switch point with a positive rate of profits. This is not the first example I present in a draft paper.

2.0 Technology

Consider the technology illustrated in Table 1. The managers of firms know of three processes of production. These processes exhibit Constant Returns to Scale. The column for the iron industry specifies the inputs needed to produce a ton of iron. Two processes are known for producing corn, and their coefficients of production are specified in the last two columns in the table. Each process is assumed to require a year to complete and uses up all of its commodity inputs. (The coefficients were "nicer" fractions before I started perturbing it. Octave code was useful.)

Table 1: The Technology

Input	Industry
	Iron	Corn
		Alpha	Beta
Labor	1	5,191/5,770	305/494
Iron	9/20	1/40	3/1976
Corn	2	1/10	229/494

This technology presents a problem of the choice of technique. The Alpha technique consists of the iron-producing process and the corn-producing process labeled Alpha. Similarly, the Beta technique consists of the iron-producing process and the corn-producing process labeled Beta.

3.0 Quantity Flows

Quantity flows can be analyzed independently of prices. Suppose the economy is in a self-replacing state, with a net output consisting only of corn. Table 2 displays quantity flows for the Alpha technique, when the net output consists of a bushel of corn. The last row shows gross outputs, for each industry. The entries in the three previous rows are found by scaling the coefficients of production in Table 1 by these gross outputs. Table 3 displays corresponding quantity flows for the Beta technique.

Consider the quantity flows for the Alpha technique. The row for iron shows that each year, the sum (9/356) + (11/356) = 5/89 tons are used as iron inputs in the iron and corn industries. These inputs are replaced at the end of the year by the output of the iron industry, with no surplus iron left over. In the corn industry, the sum 10/89 + 11/89 = 21/89 bushels are used as corn inputs in the two industries. When these inputs are replaced out of the output of the corn industry, a surplus of one bushel of corn remains. The net output of the economy, when these processes are operated in these proportions, is one bushel corn. The table allows one to calculate, for each technique, the labor aggregated over all industries per net unit output of the corn industry. Likewise, one can find the aggregate physical quantities of capital goods per net unit output of corn.

Table 2: Quantity Flows for Alpha Technique

Input	Industry
	Iron	Corn
Labor	5/89 ≈ 0.0562 Person-Yrs.	57,101/51,353 ≈ 1.11 Person-Yrs.
Iron	9/356 ≈ 0.0253 Tons	11/356 ≈ 0.0309 Tons
Corn	10/89 ≈ 0.112 Bushels	11/89 ≈ 0.124 Bushels
Output	5/89 ≈ 0.0562 Tons	110/89 ≈ 1.24 Bushels

Table 3: Quantity Flows for Beta Technique

Input	Industry
	Iron	Corn
Labor	3/577 ≈ 0.00520 Person-Yrs.	671/577 ≈ 1.16 Person-Yrs.
Iron	27/11,540 ≈ 0.00234 Tons	33/11,540 ≈ 0.00286 Tons
Corn	6/577 ≈ 0.0104 Bushels	2,519/2885 ≈ 0.873 Bushels
Output	3/577 ≈ 0.00520 Tons	5,434/2,885 ≈ 1.88 Bushels

4.0 Prices and the Choice of Technique

The choice of technique is analyzed based on prices of production and cost-minimization. Labor is assumed to be advanced, and wages are paid out of the surplus product at the end of the year. Corn is taken as the numeraire. Figure 1 graphs the wage curve for the two techniques. The cost-minimizing technique, at a given rate of profits, maximizes the wage. That is, the cost-minimizing techniques form the outer envelope, also known as, the wage frontier, from the wage curves. In the example, the Beta technique is cost minimizing for high rates of profits, while the Alpha technique is cost-minimizing between the two switch points. At the switch points, any linear combination of the two techniques is cost-minimizing.

One switch point is a fluke; it occurs for a rate of profits of zero. Any infinitesimal variation in the coefficients of production would result in the switch point no longer being on the wage axis. This intersection between the wage curves would then either occur at a negative or positive rate of profits. In the former case, the example would be one with a single switch point with a non-negative, feasible rate of profits, and the real Wicksell effect would be negative at that switch point. In the latter case, it would be a reswitching example, with the Beta technique uniquely cost-minimizing for low and high rates of profits. The real Wicksell effect would be negative at the first switch point and positive at the second.

5.0 Aggregates

In calculating wage curves, one can also find prices for each rate of profits. Table 5 shows certain aggregates, as obtained from Tables 2 and 3 and the price of iron at the switch point with a positive rate of profits. (Table 4 shows this price.) The numeraire value of capital per person-year, for a given technique and a given rate of profits, is the additive inverse of the slope of a line joining the intercept of the technique's wage curve with the wage axis to a point on the wage curve at the specified rate of profits. The capital-labor ratio, for a given technique, varies with the rate of profits, unless the wage curve is a straight line. Since a switch point occurs on the wage axis, the capital-labor ratio for both techniques at the other switch point is identical. As seen in Table 5, it does not vary among the two cost-minimizing techniques at the switch point with a positive rate of profits. The real Wicksell effect is zero at this switch point.

Table 4: Price Variables at Switch Point with Real Wicksell Effect of Zero

Variable	Value
Rate of Profits	125,483/209,727 ≈ 59.8 Percent
Wage	9,226,807/24,957,513 ≈ 0.370 Bushels per Person-Yr.
Wage	7,558/595 ≈ 12.7 Bushels per Ton

Table 5: Aggregates at Switch Point with Real Wicksell Effect of Zero

	Technique
	Alpha	Beta
Net Output	1 Bushel Corn
Labor	674/577 ≈ 1.17 Person-Years
Physical Capital	5/89 Tons Iron	3/577 Tons Iron
	21/89 Bushels Corn	2,549/2,885 Bushels Corn
Financial Capitl	113/119 ≈ 0.945 Bushels Corn
Capital-Labor Ratio	65,201/80,206 ≈ 0.813 Bushels per Person-Yr.

6.0 Implications

A certain sort of indeterminacy arises in the example. For a given quantity of corn produced net, the ratio of labor employed in corn production to labor employed in iron production varies, at the switch point with a positive rate of profits, from around 1/5 to just over 223 to one. A change in technique leaves total employment unchanged, given net output, even as it alters the allocation of labor among industries. At the switch point, a change in technique, given net output, leaves the total value of capital unchanged, while, once again, altering its allocation among industries.

Suppose the economy is in a stationary state with the wage slightly below the wage at the switch point with a real Wicksell effect of zero. The Beta technique is in use. Consider what happens if a positive shock to wages result in a wage permanently higher than the wage at the switch point. The shock might be, for example, from an unanticipated increase in the minimum wage. Prices and outputs will be out of proportion, and a perhaps long disequilibrium adjustment process begins. Suppose that, eventually, after all this folderol, the economy, once more, attains another stationary state. The Alpha technique will now be in use. Labor hired per unit net output will be unchanged. The only variation in the value of capital goods per unit labor is a result of price changes, independent of the change in technique.

Bifurcation Diagram for Fluke Switch Point

Figure 1: A Bifurcation Diagram

I have previously illustrated a case in which real Wicksell effects are zero. I wrote this post to present an argument that that example is not a matter of round-off error confusing me.

Consider the technology illustrated in Table 1. The managers of firms know of three processes of production. These processes exhibit Constant Returns to Scale. The column for the iron industry specifies the inputs needed to produce a ton of iron. Two processes are known for producing corn, and their coefficients of production are specified in the last two columns in the table. Each process is assumed to require a year to complete and uses up all of its commodity inputs.

Table 1: The Technology

Input	Industry
	Iron	Corn
		Alpha	Beta
Labor	1	a0,2α	305/494
Iron	9/20	1/40	3/1976
Corn	a2,1	1/10	229/494

Figure 1 shows two loci in the parameter space defined by the two coefficients of production a_0,2^α and a_2,1. The solid line represents coefficients of production for which the wage curves for the two techniques are tangent at a point of intersection. The dashed line represents parameters for which a switch point exists on the wage axis. The point at which these two loci are tangent specifies the parameters for this example. Figure 2 repeats the graph of the wage curves for that example.

Figure 2: A Fluke Switch Point

Suppose coefficients are as in the example in the main text, but a_0,2^α is somewhat greater. Then the wage curve for the Alpha technique lies below the wage curve for Beta for all non-negative rates of profits not exceeding the maximum rate of profits. For all feasible rate of profits, Beta is cost-minimizing. On the other hand, if a_0,2^α is somewhat less than in the example, the wage curve for Alpha is somewhat higher than in Figure 2. The wage curve for Alpha will intersect the wage curve for Beta at two points, one with a negative rate of profits exceeding one hundred percent and one for a switch point with a positive rate of profit. As indicated in Figure 1, this combination of parameters is an example of the reserve substitution of labor

In the region graphed in Figure 1, if the coefficient of production a_0,2^α falls below the loci at which the two wage curves are tangent, the wage curves will have two intersections. Suppose a_2,1 is greater than in the example in the main text. In the corresponding region between the two loci in Figure 1, the rate of profits at both intersections of the wage curves are negative. In this region of the parameter space, Beta remains cost-minimizing for all feasible non-negative rates of profits. If a_2,1 is less than in the example, the rate of profits for both intersections are positive in the region between the two loci. The example is one of reswitching. In effect, which intersection of the wage curves is a switch point on the wage axis changes along the locus for the switch point on the wage axis.

Consider the rate of profits at which the wage curves have a repeated intersection, that is, are tangent, for the corresponding locus in Figure 1. Toward the left of the figure, this rate of profits is positive, while it is negative toward the right. By continuity, this rate of profits is zero for a single point in the graphed part of the parameter space. The two loci must be tangent for this set of parameters. The appearance of a switch point with a real Wicksell effect of zero in this post is not a result of round-off error or finite precision arithmetic. Such a point exists for exactly specified coefficients of production.

Fluke Switch Points and a Real Wicksell Effect of Zero

I have put up a draft paper with the post title on my SSRN site.

Abstract: This note presents two numerical examples, in a model with two techniques of production, of a switch point with a real Wicksell effect of zero. The variation in the technique adopted, at the switch point, leaves employment and the value of capital per unit net output unchanged. This invariant generalizes to switch points with a real Wicksell effect of zero for steady states with a positive rate of growth.

A Fluke Switch Point With A Real Wicksell Effect Of Zero

Figure 1: A Fluke Switch Point

1.0 Introduction

A switch point in which the wage curves for two techniques are tangent to one another at the switch point is a fluke. Likewise, a switch point that occurs at a rate of profits of zero is a fluke. This post presents a two-commodity example with a choice between two techniques, in which the single switch point is simultaneously both types of flukes. The wage curves are tangent at the switch point, and the switch point occurs at a rate of profits of zero.

2.0 Technology

Consider the technology illustrated in Table 1. The managers of firms know of three processes of production. These processes exhibit Constant Returns to Scale. The column for the iron industry specifies the inputs needed to produce a ton of iron. Two processes are known for producing corn, and their coefficients of production are specified in the last two columns in the table. Each process is assumed to require a year to complete and uses up all of its commodity inputs. (The coefficients were found by first creating an example with two wage curves tangent at a switch point. Selected coefficients were then varied to move the switch point to the wage axis. A binary search improved the approximation. Octave code was useful.)

Table 1: The Technology

Input	Industry
	Iron	Corn
		Alpha	Beta
Labor	1	0.802403	305/494
Iron	9/20	1/40	3/1976
Corn	3.9973702	1/10	229/494

This technology presents a problem of the choice of technique. The Alpha technique consists of the iron-producing process and the corn-producing process labeled Alpha. Similarly, the Beta technique consists of the iron-producing process and the corn-producing process labeled Beta.

3.0 Quantity Flows

Quantity flows can be analyzed independently of prices. Suppose the economy is in a self-replacing state, with a net output consisting only of corn. Table 2 displays (approximate) quantity flows for the Alpha technique, when the net output consists of a bushel of corn. The last row shows gross outputs, for each industry. The entries in the three previous rows are found by scaling the coefficients of production in Table 1 for Alpha by these gross outputs. The row for iron shows that each year, the sum 0.02848 + 0.3480 = 0.6328 tons are used as inputs in the iron and corn industries. These inputs are replaced at the end of the year by the output of the iron industry, with no surplus iron left over. Similarly, the output of the corn industry replaces the inputs of corn for the two industries, leaving a net output of one bushel corn.

Table 2: Quantity Flows for Alpha Technique

Input	Industries
	Iron	Corn
Labor	0.06328	1.11708
Iron	0.02848	0.03480
Corn	0.25296	0.13922
Outputs	0.06328	1.39217

Table 3 shows corresponding quantity flows for the Beta technique. As above, the net output is one bushel corn. These tables allow one to calculate, for each technique, the labor aggregated over all industries per net unit output of the corn industry. Likewise, one can find the aggregate physical quantities of capital goods per net unit output of corn.

Table 3: Quantity Flows for Beta Technique

Input	Industries
	Iron	Corn
Labor	0.00525	1.17512
Iron	0.00236	0.00289
Corn	0.02100	0.88230
Outputs	0.00525	1.90330

4.0 Prices

The choice of technique is analyzed based on prices of production and cost-minimization. Labor is assumed to be advanced, and wages are paid out of the surplus product at the end of the year. Corn is taken as the numeraire. Figure 1 graphs the wage-rate of profits for the two techniques. The cost-minimizing technique, at a given rate of profits, maximizes the wage. That is, the cost-minimizing techniques form the outer envelope, also known as, the wage frontier, from the wage curves. The Beta technique is cost-minimizing at all feasible rates of profits. At the switch point, the Alpha technique is also cost-minimizing. Furthermore, at the switch point, any linear combination of the techniques is cost-minimizing.

In calculating wage curves, one can also find prices for each rate of profits. Table 4 shows certain aggregates, as obtained from Tables 2 and 3 and the price of iron at the switch point.

Table 4: Aggregates at the Switch Point

Aggregate	Technique
	Alpha	Beta
Net Output	1 Bushel Corn
Labor	1.18036 Person-Years
Physical Capital	0.06328 Tons 0.39217 Bushels	0.00525 Tons, 0.90330 Bushels
Financial Capital	0.94957 Bushels

A certain sort of indeterminancy arises in the example. For a given quantity of corn produced net, the ratio of labor employed in corn production to labor employed in iron production varies at the switch point from approximately 17.7 to 223.7. A change in technique leaves total employment unchanged, given net output, even as it alters the allocation of labor between industries. It is also the case that, at the switch point, a change in technique, given net output, leaves the total value of capital unchanged, while, once again, altering its allocation between industries.

For non-fluke switch points, aggregate employment and the aggregate value of capital, per unit net output, vary with the technique. If the technique that is cost minimizing at an infinitesimally greater rate of profits than associated with the switch point has a greater value of capital per net output at the switch point, the real Wicksell effect is positive. If that technique has a smaller value of capital per net output, still using the prices at the switch point to value capital goods, is negative. (Edwin Burmeister argues that a negative real Wicksell effect is the appropriate formalization of the neoclassical idea of capital-deepening.) The fluke switch point presented here has a zero real Wicksell effect.

The indeterminacy at the switch point is related to both fluke properties of the switch point. Net output per worker, for a given technique, is shown by the intersection of the wage curve for the technique with the wage axis. Since both curves intersect the wage axis at the same point, they produce the same net output per worker. Thus, both techniques result in the same overall employment, per bushel corn produced net.

The wage curve also shows the value of capital per worker. For a given technique and rate of profits, the numeraire value of capital per person-year is the absolute value of the slope of the secant connecting the point on the wage curve specified by the rate of profits and the intercept with the wage axis. In the limit, when the rate of profits is zero, the value of capital per person-year is the absolute value of the slope of the tangent. The tangency of the wage curves at the switch point on the wage axis implies that both techniques have the same value of capital per person-year.

Update (10 Sept. 2017): Fixed transcription error in coefficients of production.

Example With Four Normal Forms For Bifurcations Of Switch Points

Figure 1: A Blowup of a Bifurcation Diagram

1.0 Introduction

I have been working on an analysis of structural economic dynamics with a choice of technique. Technical progress can result in a variation in the switch points and the succession of techniques with wage curves on the outer wage frontier. I call such a variation a bifurcation, and I have identified normal forms for four generic bifurcations. This post prevents an example in which all four generic bifurcations appear.

2.0 Technology

The example in is one of an economy in which four commodities can be produced. These commodities are called iron, copper, uranium, and corn. The managers of firms know of one process for producing each of the first three commodities. They know of three processes for producing corn. Table 1 specifies the inputs required for a unit output for each of these six processes. Each column specifies the inputs needed for the process to produce a unit output of the designated industry. Variations in the parameters a_{11, β} and a_{11, γ} can result in different switch points appearing on the frontier.

Table 1: The Technology for Three of Four Industries

Input	Industry
	Iron	Copper	Uranium
Labor	1	17,328/8,281	1
Iron	1/2	0	0
Copper	0	a11, β	0
Uranium	0	0	a11, γ
Corn	0	0	0

Three processes are known for producing corn (Table 2). As usual, these processes exhibit CRS. And each column specifies inputs needed to produce a unit of corn with that process.

Table 2: The Technology the Corn Industry

Input	Process
	Alpha	Beta	Gamma
Labor	1	361/91	3.63505
Iron	3	0	0
Copper	0	1	0
Uranium	0	0	1.95561
Corn	0	0	0

3.0 Technical Progress

3.1 Progress in Copper Production

Consider the variation in the number and location of switch points as the coefficient of production for the input of copper per unit copper produced, a_{11, β}, falls from over 48/91 to around 1/4. In this analysis, the coefficient of production for the input of uranium per unit uranium produced, a_{11, γ}, is set to 3/5. This variation in a_{11, β}, while all other coefficients of production are fixed, describes a type of technical progress in the copper industry.

Figure 2 shows the configuration of wage curves near the start of this story. The Gamma technique is never cost-minimizing. For all feasible rates of profits, the wage curve for the Gamma technique falls within the wage frontier. For a parameter value of a_{11, β} of 48/91, the Alpha technique is always cost-minimizing. A single switch point exists, at which the wage curve for the Beta technique is tangent to the wage curve for the Alpha technique, and the Beta technique is also cost-minimizing. I call a configuration of wage curves like that in Figure 2 a reswitching bifurcation. For a slightly lower value of a_{11, β}, two switch points would emerge. The Alpha technique would be cost-minimizing for low and high rates of profits, with the Beta technique cost-minimizing at intermediate rates of profits.

Figure 2: A Reswitching Bifurcation

Figure 3 shows the configuration of wage curves when a_{11, β} has fallen to one half. The interval with high rates of profits where the Alpha technique is uniquely cost-minimizing has vanished. The switch point between Alpha and Beta at high rates of profits occurs at a wage of zero. I call Figure 3 an example of a bifurcation around the axis for the rate of profits. For a slightly smaller value of a_{11, β}, the switch point on the axis would vanish, and only one switch point would exist, in this example, for a non-negative wage.

Figure 3: A Bifurcation around the Axis for the Rate of Profits

Suppose the coefficient of production a_{11, β} were to fall to approximately 0.31008. Figure 4 shows the resulting configuration of wage curves. The Beta technique is cost-minimizing for all feasible positive rates of profit. A single switch point exists, between Alpha and Beta, on the wage axis. If a_{11, β} were to fall even further, no switch points would exist, and Beta would also be cost-minimizing for a rate of profits of zero. I call this an example of a bifurcation around the wage axis.

Figure 4: A Bifurcation around the Wage Axis

Figures 5 and 6 summarize the above discussion. The coefficient of production a_{11, β} is plotted on the abscissa in each figure. The rates of profits and the wage, respectively, are plotted on the ordinate. Switch points are graphed. The maximum rates of profits for the Alpha and Beta technique are plotted in Figure 5. In Figure 6, the maximum wages for Alpha and Beta are plotted. Each of the three bifurcations in Figure 2, 3, and 4 is shown as a thin vertical line in Figures 5 and 6. The wage curve for the Beta techniques moves outward as one passes from the right to the left in the figures. One can see the single switch point becoming two, and the distance between the two, in terms of either the rate of profits of the wage, becoming greater. The rate of profits for one switch point eventually exceeds the maximum rate of profits and disappears. The rate of profits for the other switch point falls below zero, leaving Beta cost-minimizing for all feasible rates of profits and wages. In short, structural economic dynamics, for the case examined here, can be summarized in either one of these two graphs.

Figure 5: A Bifurcation Diagram for Technical Progress in the Copper Industry

Figure 6: A Bifurcation Diagram for Technical Progress in the Copper Industry

3.2 Progress in Uranium Production

An analysis of technical progress in the uranium industry illustrates another type of bifurcation. Let a_{11, β} be set to 51/100, and let the coefficient of production for the input of uranium per unit uranium produced, a_{11, γ}, fall from around 0.55 to 0.4. Figure 7 shows the configuration of wage curves when a_{11, γ} is approximately 0.537986. The wage curves for Alpha and Beta exhibit reswitching. The wage curve for the Gamma technique also intersects the switch point at the lower rate of profits. I call such a configuration of wage curves a three-technique bifurcation. Aside from the switch point, the Gamma technique is never cost-minimizing.

Figure 7: A Three Technique Bifurcation

As a_{11, γ} decreases, the wage curve for the Gamma technique moves outward. At an intermediate value, the wage curve for Gamma intersects the wage curves for Alpha and Beta at different switch points. The reswitching example is transformed into one of capital reversing without reswitching.

Figure 8 displays a case where the wage curve for Gamma has moved outwards until it intersects the other switch point for the reswitching example. Other than at the switch point, the Beta technique is not cost minimizing for any feasible rate of profits. Figure 8 is also a case of a three-technique bifurcation.

Figure 8: Another Three Technique Bifurcation

Figure 9 is a bifurcation diagram illustrating this analysis of technical progress in the uranium industry. It graphs the rate of profits against the coefficient of production a_{11, γ}. Switch points on the wage frontier, as well as the maximum rates of profits for the Alpha and Gamma technique, are graphed. The two thin vertical lines toward the right side of the graph are the two three-technique bifurcations. For a slightly lower value of a_{11, γ} than used in Figure 8, this is a reswitching example between Alpha and Gamma. As a_{11, γ} falls even lower, both switch points disappear over the axis for the rate of profits and the wage, respectively, in a graph of wage curves. That is, this example exhibits another illustration of both a bifurcation around the axis for the rate of profits and a bifurcation around the wage axis.

Figure 9: A Bifurcation Diagram for Technical Progress in the Uranium Industry

3.3 Another Bifurcation Diagram

Sections 3.1 and 3.2 each graph switch points against a parameter in the numerical example. A more comprehensive analysis would consider all possible combinations of valid parameter values. One would need to draw a twelve-dimensional space. A part of the space defined by feasible combinations of positive values of a_{11, β} and a_{11, γ} is illustrated in Figure 10, instead Eleven regions are numbered in the figure. Figure 1 enlarges part of Figure 10 and labels the loci dividing regions with specific types of bifurcations.

Figure 10: A Bifurcation Diagram for the Parameter Space

Each numbered region contains an interior. For points in the interior of a region, a sufficiently small perturbation of the coefficients of production a_{11, β} and a_{11, γ} leaves unchanged the number and pattern of switch points. The sequence of cost-minimizing techniques along the wage frontier between switch points is also invariant within regions. Accordingly, Table 3 lists switch points and cost-minimizing techniques for each region. The techniques are specified in order, from a rate of profits of zero to the maximum rate of profits. In several regions, such as region 2, the same technique is listed more than once, since it appears on the wage frontier in two disjoint intervals. Each locus dividing a pair of regions is a bifurcation. The reader can check that the labels for bifurcations in Figure 1 are consistent with Table 3.

Table 3: Techniques on the Wage Frontier

Region	Techniques
1	Alpha throughout
2	Alpha, Beta, Alpha
3	Alpha, Beta
4	Beta throughout
5	Alpha, Gamma, Alpha
6	Alpha, Gamma, Alpha, Beta, Alpha
7	Alpha, Gamma, Beta, Alpha
8	Alpha, Gamma, Beta
9	Alpha, Gamma
10	Gamma
11	Gamma, Beta

To aid in visualization, Figures 11, 12, and 13 graph wage curves and switch points on the wage frontier for each of the eleven regions. Within a region, the number of and characteristics of intersections of wage curves not on the frontier can vary. For example, the graph for region 8 in the lower right of Figure 12 shows an intersection between the wage curves for the Alpha and Gamma techniques at a high rate of profits. That second intersection between these wage curves can disappear over the axis for the rate of profits while leaving the sequence, if not the location, of cost-minimizing techniques and switch points on the frontier unchanged.

Figure 11: Wage Curves for Regions 1 through 4

Figure 12: Wage Curves for Regions 5 through 8

Figure 13: Wage Curves for Regions 9 through 11

The numerical example is an instance of the Samuelson-Garegnani model. Variations in the two coefficients of production for the copper industry have no effect on the location of intersections between wage curves for Alpha and Gamma. Thus, one obtains the horizontal lines in Figures 1 and 10. Likewise, variations in a_{11, γ} do not affect intersections between the wages curves for Alpha and Beta. This property results in the vertical lines in the bifurcation diagram. Bifurcations in which wage curves for both Beta and Gamma are involved result in the more or less diagonal curves in Figures 1 and 10.

Section 3.1 tells a tale of technical progress in the copper industry. This story is illustrated by the bifurcation diagrams in Figures 1 and 10. The chosen values for a_{11, β} divide regions 1, 2, 3, and 4. Figure 2 lies along the vertical line dividing regions 1 and 2. Figure 3 illustrates the division between regions 2 and 3, and Figure 4 illustrates the corresponding division between regions 3 and 4. The vertical line towards the left side of Figure 10 is a bifurcation across the wage axis.

Similarly, Section 3.2 illustrates bifurcations along a movement downward in Figures 1 and 10. Such a downward movement would pass through regions 2, 7, 5, 9, and 10. Figure 7 illustrates parameters on the locus dividing regions 2 and 7. Figure 8 illustrates the division between regions 7 and 5. The line dividing regions 5 and 9 is a bifurcation around the axis for the rate of profits, and the line dividing regions 9 and 10 is a bifurcation around the wage axis. All four bifurcations are illustrated in Figure 9.

The above partitioning of the parameter space formed by coefficients of production suggests the existence of bifurcations not yet illustrated. For example, a three-technique bifurcation is located anywhere along the locus dividing regions 6 and 7. This bifurcation differs from the three-technique bifurcations illustrated by Figures 7 and 8. Or consider the point that separates regions 1, 2, 5, and 6. The Alpha technique is cost minimizing for all feasible rates of profits for these coefficients of production. Two switch points exist, and at each one of these switch points another technique is tied with the Alpha technique. The wage curve for the Gamma technique is tangent to the wage curve for the Alpha technique at the switch point with the lower rate of profits. The wage curve for the Beta technique is tangent to the wage curve for the alpha technique at the other switch point. The point on the intersection between the loci dividing regions 2, 6, and 7 is interesting. The coefficients of production specified by this point characterize a three-technique bifurcation in which the wage curves for the Alpha and Gamma techniques are tangent at the appropriate switch point. This discussion has not exhausted the possibilities.

The Concept Of Totality

This post is inspired by current events

"It is not the primacy of economic motives in historical explanation that constitutes the decisive difference between Marxism and bourgeois thought, but the point of view of totality. The category of totality, the all-pervasive supremacy of the whole over the parts is the essence of the method which Marx took over from Hegel and brilliantly transformed into the foundations of a wholly new science. The capitalist separation of the producer from the total process of production, the division of the process of labour into parts at the cost of the individual humanity of the worker, the atomisation of society into individuals who simply go on producing without rhyme or reason, must all have a profound influence on the thought, the science and the philosophy of capitalism. Proletarian science is revolutionary not just by virtue of its revolutionary ideas which it opposes to bourgeois society, but above all because of its method. The primacy of the category of totality is the bearer of the principle of revolution in science.

The revolutionary nature of Hegelian dialectics had often been recognised as such before Marx, notwithstanding Hegel's own conservative applications of the method. But no one had converted this knowledge into a science of revolution. It was Marx who transformed the Hegelian method into what Herzen described as the 'algebra of revolution'. It was not enough, however, to give it a materialist twist. The revolutionary principle inherent in Hegel's dialectic was able to come to the surface less because of that than because of the validity of the method itself, viz. the concept of totality, the subordination of every part to the whole unity of history and thought. In Marx the dialectical method aims at understanding society as a whole. Bourgeois thought concerns itself with objects the arise either from the process of studying phenomena in isolation, or from the division of labour and specialisation in the different disciplines. It holds abstractions to 'real' if it is naively realistic, and 'autonomous' if it is critical."

-- Georg Lukács, History and Class Consciousness (trans. by Rodney Livingstone), MIT Press (1971): pp. 27-28.