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.

A Reswitching Bifurcation, Reflected

Figure 1: Two Bifurcation Diagrams Horizontally Reflecting

1.0 Introduction

This post continues my investigation of structural economic dynamics. I am interested in how technological progress can change the analysis of the choice of technique. I have four normal forms for how switch points can appear on or disappear from the wage frontier, as a result of changes in coefficients of production. This post concentrates on what I call a reswitching bifurcation.

Each bifurcation can be described by how wages curves look around the bifurcation before, at, and after the bifurcation. I claim that, in some sense, order does not matter. For each normal form, bifurcations can exist in either order. I have proven this, for three of the bifurcations, by constructing the normal forms in both orders. This post completes the proof by exhibiting both orders for the reswitching bifurcation.

2.0 Technology

Consider the technology illustrated in Table 1. The managers of firms know of four processes of production. And these processes exhibit Constant Returns to Scale. The column for the iron industry specifies the inputs needed to produce a ton of iron. The column for the copper industry likewise specifies the inputs needed to produce a ton of copper. 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. Technology is defined in terms of two parameters, u and v. u denotes the quantity of labor needed to produce a unit iron in the iron industry. v is the quantity of labor needed to produce a unit copper.

Table 1: The Technology for a Three-Commodity Example

Input	Industry
	Iron	Copper	Corn
			Alpha	Beta
Labor	u	v	1	361/91
Iron	1/2	0	3	0
Copper	0	48/91	0	1
Corn	0	0	0	0

As usual, 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 copper-producing process and the corn-producing process labeled Beta.

3.0 Selected Configurations of Wage Curves

3.1 A Reswitching Bifurcation

Consider certain specified parameter values for the coefficients of production denoting the amount of labor needed to produce one unit of iron and one unit of copper. In particular, let u be 1, and let v be 17,328/8,281. Figure 2 graphs the wage curves for the two techniques in this case.

Figure 2: Wage Curves at the Bifurcation

I call this case a reswitching bifurcation. Like all bifurcations, it is a fluke case.

3.2 Improvements in Iron Production Around The Reswitching Bifurcation

Consider variations in u, from some parameter larger than its value in the above reswitching bifurcation to some lower value. In this part of the story, the value of v is assumed to be fixed at its value for the bifurcation. The right half of Figure 1, at the top of this post, illustrates this story.

For a high value of u, to the right of the right of Figure 1, the wage curve for Alpha is moved inside its location in Figure 2. The wage curves for the Alpha and Beta techniques intersect at two points. It is a reswitching example. A fall in u is illustrated by a movement to the left on the right side of Figure 1. The two switch points become closer and closer along the wage frontier. The reswitching bifurcation is illustrated by the thin vertical line in Figure 1. For any smaller value of u, the Alpha technique is cost minimizing for all feasible rates of profits or wages.

3.3 Improvements in Copper Production Around The Reswitching Bifurcation

Now consider variations in v, with u fixed at the value for the bifurcation illustrated in Figure 2. Technical progress in the copper industry is illustrated by a movement to the left on the left side of Figure 1. For a high value of v, the wage curve for the Beta technique is inside the wage curve for the Alpha technique. The Alpha technique is cost-minimizing for all feasible rates of profits. As v decreases, the wage curve for the Beta technique moves outward, until it reaches the reswitching bifurcation. For smaller values of v, the example becomes, once again, a reswitching example. A second bifurcation is illustrated on the left side of Figure 1, when the switch point at the higher rate of profits moves across the axis for the wage. The labor input for copper has become so small that the Beta technique is cost-minimizing for any sufficiently large enough wage and small rate of profits.

4.0 Conclusion

The bifurcation depends on a certain relative configuration of wage curves, in which one is tangent to the other at a switch point. Whether technical progress around the bifurcation results in reswitching appearing or disappearing depends on which wage curve is moving outwards faster around the switch point(s). Either order is possible.

Elsewhere

• Nick Hanauer argues for some policies that postulate:
• Income distribution is not a matter of supply and demand or any other sort of economic natural laws.
• That a more egalitarian distribution of income leads to an increased demand and generalized shared prosperity.
• Tom Palley contrasts neoliberalism with an economic theory with an approach with another "theory of income distribution and its theory of aggregate employment determination".
• Elizabeth Bruenig contrasts liberalism with the the left.
• Paul Blest laughs at whining neoliberals
• Chris Lehmann considers how the turn of the US's Democratic Party to neoliberalism lowers its electoral prospects.

Is the distinction between democratic socialism and social democracy of no practical importance at the moment in any nation's politics? I think of the difference in two ways. First, in the United States in the 1970s, leftists had an argument. Self-defined social democrats became Neoconservatives, while democratic socialists found the Democratic Socialists of America (DSA). Second, both are reformists approaches to capitalism, advocating tweaks to, as Karl Popper argued for, prevent unnecessary pain. But social democrats have no ultimate goal of replacing capitalism, while democratic socialists want to end up with a transformed system.

A Fluke Of A Fluke Switch Point

Figure 1: Wage Curves

1.0 Introduction

This post presents an example of the analysis of the choice of technique in competitive markets. The example is one with three techniques and two switch points. The wage curves for the Alpha and Beta techniques are tangent at one of the switch points. This is a fluke. And the wage curves for all three techniques all pass through that same switch point. This, too, is a fluke.

I suppose that the example is one of reswitching and capital-reversing is the least interesting property of the example. Paul Samuelson was simply wrong in labeling such phenomena as perverse. A non-generic bifurcation, like the illustrated one, falls out of a comprehensive analysis of possible configurations of wage curves.

2.0 Technology

The technology in the example has a particularly simple structure. Firms can produce one of three capital goods, which I am arbitrarily labeling iron, copper, and uranium. Table 1 shows the production processes known for producing each metal. One process is known for producing each, and each metal is produced out of inputs of labor and that metal. Each process requires a year to complete, uses up all its material inputs, and exhibits Constant Returns to Scale.

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	48/91	0
Uranium	0	0	0.53939
Corn	0	0	0

Three processes are known for producing corn (Table 2), which is the consumption good. This economy can be sustained by adopting one of three techniques. The Alpha technique consists of the iron-producing process and the corn-producing process labeled Alpha. Similarly, the Beta technique consists of the copper-producing process and the corn-producing process labeled Beta. Finally, the Gamma technique consists of the remaining two processes.

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 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-rate of profits for the three 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. Aside from switch points, the Alpha technique is cost-minimizing at low and high rates of profits, with the Gamma technique cost-minimizing between the switch points. At switch points, any linear combination of the techniques with wage curves going through that switch point are cost-minimizing.

The wage curve for the Beta technique is a straight line. This affine property results from the Organic Composition of Capital being the same in copper production and in corn production, when the Beta technique is adopted. To help visualization, I also graph the difference between the wage curves (Figure 2). The Beta technique is only cost-minimizing at the switch point at the higher rate of profits. The tangency of the wage curves for the Alpha and Beta techniques is manifested in Figure 2 by the non-negativity of the difference in these curves.

Figure 2: Distance Between Wage Curves

4. Conclusion

I'm sort of proud of this example. I suppose I could, at least, submit it for publication somewhere. But it is only a side effect of a larger project I guess I am pursuing.

I want to introduce a distinction among fluke switch points. Every bifurcation (that is, a change in the sequence of switch points and cost-minimizing techniques along the wage frontier) is a fluke. Some perturbation of a coefficient of production from a bifurcation value will change that sequence. Suppose a perturbation of a coefficient of production not involved in a bifurcation, in some sense, leaves the qualitative story unchanged. One can use the same bifurcation to tell a story about, say, technological progress. This is a generic bifurcation.

Accept, for the sake of argument, that prices of production tell us something about actual prices. The economy is never in an equilibrium, but owners of firms are always interested in increasing their profits. One can never expect observed technology to meet the fluke conditions of a generic bifurcation. But it can tell us something about how the dynamics of income distribution, for example, vary with technological progress.

Suppose one perturbs, in the example, the coefficient of production for the amount of iron needed to produce iron. (I denote this coefficient, in a fairly standard notation, as a_1,1^β.) Then, either the wage curves for the. Alpha and Beta techniques will not intersect at all or they will intersect twice. In the latter case, one can vary a_1,1^γ to find an example in which all three wage curves intersect at one or another of the switch points. But the tangency will be lost. So I consider the fluke point illustrated to be a non-generic bifurcation.

Non-generic bifurcations arise in a complete bifurcation analysis. The model illustrated remains open. Income distribution is not specified. Nevertheless, I think this theoretical analysis can say something to those who are attempting to empirically apply the Leontief-Sraffa model.

Some Unresolved Issues In Multiple Interest Rate Analysis

1.0 Introduction

Come October, as I understand it, the Review of Political Economy will publish, in hardcopy, my article The Choice of Technique with Multiple and Complex Interest Rates. I discuss in this post questions I do not understand.

2.0 Non-Standard Investments and Fixed Capital

Consider a point-input, flow-output model. In the first year, unassisted labor produces a long-lived machine. In successive years, labor and a machine of a specific history are used to produce outputs of a consumption good and a one-year older machine. The efficiency of the machine may vary over the course of its physical lifetime. When the machine should be junked is a choice variable in some economic models.

I am aware that in this, or closely related models, the price of a machine of a specific date can be negative. The total value of outputs at such a year in which the price of the machine of the machine is negative, however, is the difference between the sum of the price of the machine & the consumption good and the price of any inputs, like labor, that are hired in that year. Can one create a numerical example of such a case in which the net value, in a given year, is negative, where that negative value is preceded and followed by years with a positive net value?

If so, this would an example of a non-standard investment. A standard investment is one in which all negative cash flows precede all positive cash flows. In a non-standard investment at least one positive cash flow precedes a negative cash flow, and vice-versa. Non-standard investments create the possibility that all roots of the polynomial used to define the Internal Rate of Return (IRR) are complex. Can one create an example with fixed capital or, more generally, joint production in which this possibility arises?

Does corporate finance theory reach the same conclusions about the economic life of a machine as Sraffian analysis in such a case? Can one express Net Present Value (NPV) as a function combining the difference between the interest rate and each IRR in this case, even though all IRRs are complex? (I call such a function an Osborne expression for the NPV.)

3.0 Generalizing the Composite Interest Rate to the Production of Commodities by Means of Commodities

In my article, I follow Michael Osborne in deriving what he calls a composite interest rate, that combines all roots of the polynomial defining the IRR. I disagree with him, in that I do not think this composite interest rate is useful in analyzing the choice of technique. But we both obtain, in a flow-input, point output model, an equation I find interesting.

This equation states that the difference between the labor commanded by a commodity and the labor embodied in that commodity is the product of the first input of labor per unit output and the composite interest rate. Can you give an intuitive, theoretical explanation of this result? (I am aware that Osborne and Davidson give an explanation, that I can sort of understand when concentrating, in terms of the Austrian average period of production.)

A model of the production of commodities by means of commodities can be approximated by a model of a finite sequence of labor inputs. The model becomes exact as the number of dated labor inputs increases without bound. In the limit, the labor command by each commodity is a finite value. So is the labor embodied. And the quantity for the first labor input decrease to zero. Thus, the composite interest rate increases without bound. How, then, can the concept of the composite interest rate be extended to a model of the production of commodities by means of commodities?

4.0 Further Comments on Multiple Interest Rates with the Production of Commodities by Means of Commodities

In models of the production of commodities by means of commodities, various polynomials arise in which one root is the rate of profits. I have considered, for example, the characteristic equation for a certain matrix related to real wages, labor inputs, and the Leontief input-output matrix associated with a technique of production. Are all roots of such polynomials useful for some analysis? How so?

Luigi Pasinetti, in the context of a theory of Structural Economic Dynamics, has described what he calls the natural system. In the price system associated with the natural system, multiple interest rates arise, one for each produced commodity. Can these multiple interest rates be connected to Osborne's natural multiple interest rates?

5.0 Conclusion

I would not mind reading attempts to answer the above questions.

Switch Points and Normal Forms for Bifurcations

I have put up a working paper, with the post title, on my Social Sciences Research Network (SSRN) site.

Abstract: The choice of technique can be analyzed, in a circulating capital model of prices of production, by constructing the wage frontier. Switch points arise when more than one technique is cost-minimizing for a specified rate of profits. This article defines four normal forms for structural bifurcations, in which the number and sequence of switch points varies with a variation in one model parameter, such as a coefficient of production. The 'perversity' of switch points that appear on and disappear from the wage frontier is analyzed. The conjecture is made that no other normal forms exist of codimension one.

Switch Points Disappearing Or Appearing Over The Axis For The Rate Of Profits

Figure 1: Two Bifurcation Diagrams Horizontally Reflecting

1.0 Introduction

This post continues my investigation of structural economic dynamics. I am interested in how technological progress can change the analysis of the choice of technique. In this case, I explore how a decrease in a coefficient of production can cause a switch point to appear or disappear over the axis for the rate of profits.

2.0 Technology

Consider the technology illustrated in Table 1. The managers of firms know of four processes of production. And these processes exhibit Constant Returns to Scale. The column for the iron industry specifies the inputs needed to produce a ton of iron. The column for the copper industry likewise specifies the inputs needed to produce a ton of copper. 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. Technology is defined in terms of two parameters, u and v. u denotes the quantity of iron needed to produce a unit iron in the iron industry. v is the quantity of copper needed to produce a unit copper.

Table 1: The Technology for a Three-Commodity Example

Input	Industry
	Iron	Copper	Corn
			Alpha	Beta
Labor	1	2/3	1	2/3
Iron	u	0	1/3	0
Copper	0	v	0	1/3
Corn	0	0	0	0

As usual, 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 copper-producing process and the corn-producing process labeled Beta.

I make all my standard assumptions. The choice of technique is based on cost-minimization. Consider prices of production, which are stationary prices that allow the economy to be reproduced. A wage curve, showing a tradeoff between wages and the rate of profits, is associated with each technique. In drawing such a curve, I assume that a bushel corn is the numeraire and that labor is advanced. Hence, wages are paid out of the surplus product at the end of the year. The chosen technique, for, say, a given wage, maximizes the rate of profits. The wage curve for the cost-minizing technique at the given wage lies on the outer frontier formed from all wage curves.

3.0 Innovations

I have two stories of technical innovation. In one, improvements are made in the process for producing copper. As a consequence, the wage curve for the Beta technique moves outward. In the other story, improvements are made in the iron industry, and the wage curve for the Alpha technique moves outwards. The bifurcations that occur in the two stories are mirror reflections of one another, in some sense.

3.1 Improvements in Copper Production

Let u be fixed at 1/3 tons per ton. The wage curve for the Alpha technique is a downward sloping straight line. Let v decrease from 1/2 to 3/10. When v is 1/3, the wage curve for the Beta technique is also a straight line. I created the example to have linear (actually, affine) wage curves at the bifurcation for convenience. The bifurcation does not require such.

Figure 2 shows the wage curves when the copper coefficient for copper production is a high value, in the range under consideration. A single switch point exists, and the Alpha technique is cost-minimizing if the rate of profits is high. As v decreases, the switch point moves to a higher and higher rate of profits. (These statements are about the shapes of mathematical functions. They are not about historical processes set in time.) Figure 3 shows the wage curves when v is 1/3. The switch point is now on the axis for the rate of profits. For any non-negative rate of profits below the maximum, the Beta technique is cost-minimizing. Finally, Figure 4 shows the wage curves for an even lower copper coefficient in copper production. Now, there is no switch point, and the Beta technique is always cost-minimizing, for all possible prices of production.

Figure 2: Wage Curves Without Improvement in Copper Production

Figure 3: Wage Curves For A Bifurcation

Figure 4: Wage Curves After Improvements in Copper Production

3.2 Improvements in Iron Production

Now let v be set at 1/3. Let u decrease from 1/2 to 3/10. Figure 5 shows the wage curves at the high end for the iron coefficient in iron production. No switch point exists, and the Beta technique is always cost-minimizing. I thought about repeating Figure 3, for v decreased to 1/3. The same configuration of wage curves, with a bifurcation, appears in this story. Figure 6, shows that the switch point appears for an even lower value of the iron coefficient.

Figure 5: Wage Curves Without Improvement in Iron Production

Figure 6: Wage Curves After Improvements in Iron Production

3.3 Improvements in Both Iron and Copper Industries

I might as well graph (Figure 7) the copper coefficient in copper production against the iron coefficient in iron production. The bifurcation occurs when the maximum rates of profits are identical in the Alpha and Beta technique. In a model with the simple structure of the example, this occurs when u = v. Representative illustrations of wage curves are shown in the regions in the parameter space. A switch point below the maximum rate of profits exists only above the line in parameter space representing the bifurcation.

Figure 7: Bifurcation Diagram for Two Coefficients of Production

The story in Section 3.1 corresponds to moving downwards on a vertical line in Figure 7. The left-hand side of Figure 1, at the top of this post, is another way of illustrating this story. On the other hand, Section 3.2 tells a story of moving leftwards on a horizontal line in Figure 7. The right-hand side of Figure 1 illustrates this story.

Focus on the intersections, in the two sides of Figure 1 of the blue, red, and purple loci. Can you see that, in some sense, they are reflections, up to a topological equivalence?

4.0 Discussion

I have a reswitching example with a switch point disappearing over the axis for the rate of profits. In that example, the disappearing switch point is 'perverse', that is, it has a positive real Wicksell effect. In the examples in Section 3 above, the disappearing or appearing switch point is 'normal', with a negative real Wicksell effect.