Tuesday, December 12, 2017

An Example of Bifurcation Analysis with Land and the Choice of Technique

Figure 1: A Bifurcation Diagram
1.0 Introduction

I have been looking at how bifurcation analysis can be applied to the choice of technique in models in which all capital is circulating capital. In my sense, a bifurcation occurs when a switch point appears or disappears off the wage frontier. A question arises for me about how to apply or visualize bifurcations in models with land, fixed capital, and so on.

This post starts to investigate this question by looking at a numerical example of a overly simple model with land and extensive rent.

2.0 Parameters and Assumptions for the Model

Table 1 specifies the technology for this example. One parameter, the labor coefficient a0β, is left free. Managers of firms know of two processes for producing corn from inputs of labor, (a type of) land, and seed corn. Each process is defined in terms of coefficients of production. All processes exhibit constant returns to scale; require a year to complete; and totally use up, in producing their output, the capital good required as input. Land, of the specified type, exits the production process as good as it was at the start of the year.

Table 1: Processes For Producing Corn
InputCorn Industry
AlphaBeta
Labora0α = 1 Person-Yr.a0β Person-Yr.
Landbα = 10 Acres of Type Ibβ = 20 Acres of Type II
Cornaα = (1/4) Bushelsaβ = (1/5) Bushels

Each type of land is in fixed supply:

  • LI = 100 Acres of Type I land exist.
  • LII = 100 Acres of Type II land exist.

The assumptions so far impose some limits on the quantity of net output that can be produced. If only Type I land is seeded, and that land is fully used, net output consists of:

(1 - aα) LI/bα = (15/2) bushels

Likewise, if only Type II land is seeded, net output consists of 4 bushels. If net output exceeds (15/2) bushels (that is, the maximum of 15/2 and 4 bushels), both types of land will need to be seeded. If net output is less than (23/2) bushels (that is, the sum of 15/2 and 4 bushels), at least one type of land will not be fully used. Accordingly, assume:

(15/2) bushels < y < (23/2) bushels

where y is net output. Under these assumptions, one type of land is in excess supply and pays no rent.

I consider prices of production to determine rent and to find out which land is free. Since net output is taken here as a constant, no matter how much a0β may fall, I am assuming increased productivity (per worker) is taken in the form of decreased employment.

3.0 Price Equations

I take corn to be numeraire, and I assume rent and wages are paid out of the surplus at the end of the period. Prices of production must satisfy the following system of equations:

(1/4)(1 + r) + 10 ρI + w = 1
(1/5)(1 + r) + 20 ρII + a0β w = 1

where r, w, ρI, and ρII are the rate of profits, the wage, the rent on Type I land, and the rent of Type II land. All four of these distribution variables are assumed to be non-negative. The condition that at least one type of land pays a rent of zero is expressed by a third equation:

ρI ρII = 0

4.0 The Choice of Technique

I consider three solutions of the price equation, each for a different parameter value of a0β.

4.1 First Example

First, suppose a0β is (6/5) person-years per bushel. Each process yields a wage curve, under the assumption that the corresponding type of land pays no rent. Figure 1 graphs both wage curves. A simple generalization of this model would be to multiple produced commodities, with land only used in one industry. Each process in that industry would be associated with a technique, and the associated wage curve could be of any convexity, with the convexity possibly varying throughout its extent.

Figure 2: Each Type of Land Sometimes Pays Rent

In this example, in which both types of land must be used to produce the given net output, the relevant frontier is the inner frontier, shown as a solid black line in the figure. This, too, does not generalize to a multi-commodity model with more types of land. In that case, one would work from the outer frontier inward until the successive types of land could produce, at least, the given net output. This order might depend on whether the wage or the rate of profits was taken as given. Or perhaps some other theory of distribution could be analyzed.

Anyways, the type of land associated with the technique on the inner frontier, in this example, pays no rent. For low rates of profits or high wages, Type II land pays no rent. For high rates of profits or low wages, Type I land pays no rent. At the switch point, both types of land pay no rent. If the wage were given, rent on the type of land associated with the process further from the origin would come out of the super profits that would otherwise be earned on that process. If the rate of profits were given, one might see a conflict between workers and landlords. This analysis is a matter of competitive markets, inasmuch as capitalists can move their investments among industries and processes.

4.2 Bifurcation Over Wage Axis

I next consider a parameter value for a0β of (16/15) person-years per bushel. As shown in Figure 2, this is a case of a bifurcation over the wage axis. You cannot see the wage curve for the Alpha technique in the figure because it is always on the inner frontier. For any distribution of the surplus, Type I land pays no rent. If the rate of profits is zero, Type II land also pays no rent. For any positive rate of profits, landlords obtain a rent on Type II rent.

Figure 3: A Bifurcation Over the Wage Axis

4.3 Type II Land Always Pays Rent

For a final case, let a0β be one person-years per bushel. The wage curve for the Alpha technique has now rotated downwards counter clockwise so far that it never intersects the wage curve for the Beta technique. Whatever the distribution, Type I land pays no rent, and owners of Type II land receive a rent.

Figure 4: Wage Curves Never Intersect

4.4 Bifurcation Diagram

So this simple example can be illustrated with a bifurcation diagram, as seen at the top of this post. The rate of profits for the switch point is"

rswitch = (15 a0β - 16)/(5 a0β - 4)

This function asymptotically approaches the maximum rate of profits for the Alpha technique as a0β increases without bound. The wage curve for Alpha continues to become steeper and steeper. I suppose wage for the switch point approaches the wage on the wage curve for the Beta technique when the rate of profits is 300 percent.

One can also solve for the rents. When the rent on Type I land is non-negative, it is:

ρI = [(15 a0β - 16) + (4 - 5 a0β)r]/(200 5 a0β)

When the rent on Type II land is non-negative, it is:

ρII = [(16 - 15 a0β) + (5 a0β - 4)]/400

5.0 Conclusions

I am partly interested in bifurcation analysis because one can draw neat graphs to visualize the economics. For the numerical example, I would like to be able to draw three-dimensional diagrams. Imagine an axis coming out of the page for the bifurcation digram at the top of this post. I then could have a surface where the rent on one of the types of land is graphed against the rate of profits and the coefficient of production being varied parametrically.

It seems like all four of the normal forms for bifurcations of co-dimension one that I have defined may arise in examples of extensive rent. These are a bifurcation over the wage axis, a bifurcation over the axis for the rate of profits, a three-technique bifurcation, and a restitching bifurcation. They will not necessarily be on the outer frontier, however.

I think another type of bifurcation may be possible. Suppose productivity increases because coefficients of production decreases for land inputs or inputs of capital goods. Given net output, could such an increase in productivity result in some type of land that formerly paid rent (for some range of the rate of profits) becoming rent-free? Could all types of land become non-scarce? How would this sort of bifurcation look on an appropriate bifurcation diagram? Would the distinction between the order of rentability and efficiency be reflected in bifurcation analysis? Can I draw a bifurcation diagram with a discontinuity?

Thursday, December 07, 2017

Infinite Number of Techniques, One Linear Wage Curves

Coefficients for First Column in Leontief Input-Output Matrix

I have uploaded a draft paper with the post title to my SSRN site.

Abstract:This note demonstrates that the special case condition, needed for a simple labor theory of value (LTV), of equal organic compositions of capital does not suffice to determine technology. A model of the production of commodities, with circulating capital and all commodities basic, is analyzed. Given direct labor coefficients and labor values, an uncountably infinite number of Leontief input-output matrices yield the same wage curve under the conditions in which prices of production are proportional to labor values.

This paper is an update of a previous draft paper. I have posed the problem better that I am addressing, have deleted an error in my previously most general formulation, replaced the numerical example by algebra, and shortened my paper. I hope I am not restating something that I did not absorb decades ago in reading John Roemer or Michio Morishima. As of today, I think I am subjectively original.

Wednesday, November 29, 2017

Bifurcation Analysis of a Two-Commodity, Three-Technique Technology

Figure 1: A Bifurcation Diagram

This post expands on this previous post. The technology is the same, but the rates of decrease of the coefficients of production in the Beta and Gamma corn-producing processes are not fixed. Instead, I consider the full range of parameter values. (I find the graphs produced by bifurcation analysis interesting for this case, but I think a two-commodity example can be found with more pleasing diagrams.)

Anyways, Figure 1 shows a bifurcation diagram for the parameter space in this example. The region numbered 8 is not visible on the graph. Accordingly, Figure 2 below shows a much expanded picture of the parameter space around that region. The specific parameter values in the previous post lead to a temporal path along the dashed ray extending from the origin in Figure 1. (The numbering of regions in this post and the previous post do not correspond.) Although it is not obvious, the locus of points bifucating regions 9 and 10 eventually, somewhere to the right of the region shown in Figure 1 eventually decreases in slope and intercepts the dashed ray.

Figure 2: Blowup of a Part of the Bifurcation Diagram

As usual, each numbered region corresponds to a definite sequence of cost-minimizing techniques contributing wage curves along the wage frontier. Table 1 lists this sequence for each region. Some notes on switch points are provided. A switch point is called "normal" merely if it conforms to outdated neoclassical intuition. In other words, such a switch point exhibits negative real Wicksell effects. In the example, regions also exist where switch points exhibit positive real Wicksell effects.

Table 1: Cost-Minimizing Techniques by Region
RegionCost-Minimizing
Techniques
Notes
1AlphaOne technique cost-minimizing.
2Alpha, Beta"Normal" switch point.
3Beta, Alpha, BetaReswitching. Switch pt. at
highest r is "perverse".
4BetaOne technique cost-minimizing.
5Alpha, Gamma"Normal" switch point.
6Alpha, Gamma, Beta"Normal" switch points.
7Beta, Alpha,
Gamma, Beta
Recurrence of techniques.
Switch pt. at highest r is
"perverse".
8Beta, Alpha, Beta,
Gamma, Beta
Two reswitchings, two
"perverse" switch pts.
9GammaOne technique cost-minimizing.
10Gamma, Beta"Normal" switch point.
11Beta, Gamma, BetaReswitching. Switch pt. at
highest r is "perverse".

One can compare and contrast the above bifurcation diagram with the one in this post. The latter bifurcation diagram is for a specific instance of the Samuelson-Garegnani model, in which the basic commodity varies among techniques. (I have a more recent write-up of that bifurcation analysis linked to here.)

Saturday, November 25, 2017

Reswitching Without a Reswitching Bifurcation

Figure 1: A Bifurcation Diagram

This post presents another example of bifurcation analysis applied to structural economic dynamics with a choice of technique. This example illustrates:

  • Two reswitching examples appear and disappear without a reswitching bifurcation ever occurring, at least on the wage frontier.
  • Two bifurcations over the wage axis arise. At the time each bifurcation of this type occurs, another switch point for the same techniques exhibits a real Wicksell effect of zero. Thus, for each, a switch point transitions from being a "normal" switch point to a "perverse" one exhibiting capital-reversing.
  • Each of the four types of bifurcations of co-dimension one that I have identified have no preferred temporal order. For example, a bifurcation over the wage axis can add a switch point to the wage frontier. And another such bifurcation can remove a switch point, as time advances.
  • The maximum rate of profits approaches an asymptote from below as time increase without bound.

Table 1 specifies the technology for this example, in terms of two parameters, σ and φ. Managers of firms know of one process for producing iron and of three processes for producing corn. Each process is defined in terms of coefficients of production, which specify the quantities of labor, iron, and corn needed to produce a unit output for that process. All processes exhibit constant returns to scale; require a year to complete; and totally use up, in producing their output, the capital goods required as input. I consider the special case in which the rate of decrease of the coefficients of production in the Beta corn-producing process, σ, is 5 percent, and the rate of decrease of coefficients in the Gamma corn-producing process, φ, is 10 percent.

Table 1: Processes For Producing Iron and Corn
InputIron
Industry
Corn Industry
AlphaBetaGamma
Labor10.899650.71733 et1.28237 et
Iron0.450.0250.00176 et0.03375 et
Corn20.10.53858 et0.13499 et

Three techniques are available for producing a net output of, say, corn, while reproducing the capital goods used as input. The Alpha process consists of the iron-producing process and the corn-producing process labeled Alpha. And so on for the Beta and Gamma techniques.

The choice of technique is analyzed in the usual way. I assume that labor is advanced, and wages are paid out of the surplus product at the end of the year. Corn is taken as numeraire. A wage curve can be drawn for each technique, given the coefficients of production prevailing at a given moment in time. Figure 1 illustrates a case of the recurrence of techniques in the example. The cost-minimizing technique is found by constructing the outer frontier of the wage curves. In Figure 2, the cost-minimizing techniques are Beta, Alpha, Gamma, and Beta, in that order. The switch point at approximately 57 percent exhibits capital-reversing. Around the switch point, a higher wage is associated with the adoption of a more labor-intensive technique. If prices of production prevail, firms will find it cost-minimizing to hire more workers at a higher wage, given net output.

Figure 2: Wage Curves in Region 4

Figure 3 illustrates the analysis of the choice of technique for all time. Switch points along the frontier and the maximum rate of profits are plotted versus time. Figure 1, at the top of this post, is a blowup of Figure 3 from time zero to a time of five years. These pictures show which technique is cost-minimizing at each rate of profits, at each moment in time. Bifurcations are also shown. Table 2 lists the cost-minimizing techniques in each region between the bifurcations.

Figure 3: An Extended Bifurcation Diagram

Table 2: Cost-Minimizing Techniques by Region
RegionCost-Minimizing
Techniques
Notes
1AlphaOne technique cost-minimizing.
2Alpha, Beta"Normal" switch point.
3Beta, Alpha, BetaReswitching. Switch pt. at
highest r is "perverse".
4Beta, Alpha, Gamma, BetaRecurrence of techniques. Switch
pt. at highest r is "perverse".
5Beta, Gamma, BetaReswitching. Switch pt. at
highest r is "perverse".
6Gamma, Beta"Normal" switch point.
7GammaOne technique cost-minimizing.
Maximum r approaches an
asymptote.

I suppose I can extend this example to partition the complete parameter space, as in this example, with an updated write-up here. That analysis will demonstrate, by example, that this sort of bifurcation analysis applies to cases in which multiple commodities are basic in multiple techniques. It is not confined to the special case of the Samuelson-Garegnani model. I am also thinking that I could perform a bifurcation analysis where parameters that vary include the ratio of the rates of profits in various industries, as in these examples of a model of oligopoly. Maybe such an analysis will yield an empirically relevant tale of the evolution of economic duality (also known as segmented markets).

Wednesday, November 22, 2017

Bifurcation Analysis Applied to Structural Economic Dynamics with a Choice of Technique

Variation of Switch Points with Technical Progress in Two Industries

I have a new working paper - basically an update of one I have previously described.

Abstract: This article illustrates the application of bifurcation analysis to structural economic dynamics with a choice of technique. A numerical example of the Samuelson-Garegnani model is presented in which technical progress is introduced. Examples of temporal paths through the parameter space illustrate variations of the wage frontier. A single technique is initially uniquely cost-minimizing for all feasible rates of profits. Eventually, the technique for which coefficients of production decrease at the fastest rate is always cost-minimizing. This example illustrates possible variations in the existence of Sraffa effects, which arise during the transition between these positions.

Thursday, November 16, 2017

Two Techniques, One Linear Wage Curve

Coefficients for Iron-Production in the Leontief Input-Output Matrix

I have uploaded a working paper with the post title.

Abstract: This note demonstrates that the special case condition, needed for a simple labor theory of value, of equal organic compositions of capital does not suffice to determine technology. Prices do not vary across techniques for both techniques in a numeric example of a two-commodity linear model of production, and they are proportional to labor values. Both techniques yield the same wage curve, in which the wage is an affine function of the rate of profits. This indeterminancy generalizes to models with more than two produced commodities.

Friday, November 10, 2017

An Example With Two Fluke Switch Points

Figure 1: Fluke Switch Points on Each Axis
1.0 Introduction

I have developed an approach for finding examples in which either two fluke switch points exist on the wage frontier or a switch point is a fluke in more than one way. This post presents a numerical example with two fluke switch points on the frontier. Not all examples generated by this approach are necessarily interesting, although I find the approach of interest. I don't think the example in this approach is all that fascinating. I had thought that examples of real Wicksell effects of zero were somewhat interesting, but I have received disagreement.

Anyways, what I have been doing is drawing bifurcation diagrams for examples in which coefficients of production vary. The bifurcation diagram partitions a parameter space into regions in which the sequence of switch points does not vary, even though their specific locations on the wage frontier may. The loci dividing regions with topologically equivalent wage frontiers specify fluke cases. A point in the parameter space in which more than one such loci intersect specifies an example which is a fluke in more than one way.

2.0 Technology

The example is a numerical instantiation of the Samuelson-Garegnani model. A single consumption good, corn is produced from inputs of corn and one of three capital goods. Table 1 lists the coefficients of production for production processes for producing corn. Each production process in this example requires a year to complete and exhibits Constant Returns to Scale. A column in Table 1 lists the physical inputs for that process required per unit corn produced at the end of the year. Workers labor over the course of the year, and the inputs of the capital good are totally used up in the process. Managers of firms also know of a process for producing each capital good (Table 2). For a given capital good, the process for producing it requires inputs of labor and the services of that capital good.

Table 1: Processes For Producing Corn
InputCorn Industry
AlphaBetaGamma
Labor13.691743.33574
Iron301
Copper00.928500
Uranium001.79455
Corn000

Table 2: Processes For Manufacturing Capital Goods
InputIndustry
IronCopperUranium
Labor11.942900.917647
Iron0.501
Copper00.50
Uranium000.550588
Corn000

Any one of three techniques can be adopted to sustainably produce corn. The Alpha technique consists of the iron-producing process and the corresponding, labelled process for producing corn. The Beta technique consists of the copper-producing process and corresponding for producing corn. And similarly for the Gamma technique.

3.0 Prices and the Wage Frontier

For each technique, a system of two equations arises. I take corn as the numeraire and assume that labor is paid out of the surplus at the end of the year. The equations show the same rate of profits being earned for both processes comprising a technique. Given an externally specified rate of profits, the equations are a linear system. They can be solved for the wage and the price of the capital good, as functions of the rate of profits. For the wage, this function is known as the wage curve. All three wage curves, one for each technique, are graphed in Figure 1 above.

The wage frontier consists of the outer envelope of all wage curves. The curve(s) on the frontier at a given rate of profits correspond(s) to the cost-minimizing technique(s) at that rate of profits. The Gamma technique is cost-minimizing at low rates of profits, and the Beta technique is cost-minimizing at high rates of profits. These two techniques are tied - that is, both cost-minimizing - at the switch point dividing these two regions of the rate of profits.

The Alpha technique is only cost-minimizing at the switch points on the wage axis and on the axis for the rate of profits. And, it is tied, with the Gamma and Beta techniques, respectively, at these switch points. A switch point appearing on the wage axis or the axis for the rate of profits is a fluke case. So both switch points with the Alpha technique are flukes. Having Alpha participate in two fluke switch points is even more of a fluke case. For what it is worth, the fluke switch point on the axis for the rate of profits exhibits capital-reversing.

Saturday, October 28, 2017

Braess' Paradox

Figure 1: An Example Of Braess' Paradox

Braess' paradox arises in transport economics, a field for applied research in economics. I was inspired by the example in Fujishige et al. (2017) for the example in this post. Under Braess' paradox, an improvement to a transport network, and thus an increase in the number of choices available to users of the network, results in decrease performance. In reliability engineering, one says such a transport network is not a coherent system.

A transport network, for a single mode (for example, air, rail, road, or water) can be specified by:

  • A network, where a network consists of links between nodes. Links can be one-way or two way.
  • A cost for traversing each link. The cost can be a function of the demand (that is, the amount of traffic traversing that link). Cost can have a stochastic component, such as a (perceived) standard deviation for the distribution of the time to traverse a link.
  • Demands on the network, as specified by source nodes for users and the destination of each user.
  • Objective functions for the users, such as the minimization of trip time or the maximization of the probability that total trip time will not exceed a given maximization. The probability for the latter objective function is known as trip reliability.

In my example (Figure 1), two road networks are specified. The network on the right differs from the one on the left in that an additional road, between nodes A and B has been added. All links are two ways. The cost for each link is specified as the number of minutes needed to travel across the link, where two links have a cost that depends on the traffic, thus modeling the effect of congestion. The parameters XSA and XSA denote the number of vehicles traversing the respective links. Thus, the number of minutes to travel across these links is proportional to the amount of traffic, with a proportionality constant of unity. The demand is assumed to be unchanged by the addition of the new link. One hundred users want to drive their vehicles from the source node S to the node destination node D. Each driver wants to minimize their total trip time.

Table 1: Costs for Each Link
LinkCost
SAXSA Minutes
SB110 Minutes
AD110 Minutes
BDXBD Minutes
ABEither infinity or 5 Minutes

Each user has a choice of two routes, ignoring purposeless cycles, in the network on the left. These routes pass through nodes S, A, and D, or through nodes S, B, and D. The addition of the "short-cut" provides two additional routes, through nodes S, A, B, and D, and through nodes S, B, A, and D.

My method of analysis is an equilibrium assignment of users to routes. John G. Waldrop created this notion of equilibrium, as I understand it. It is an application of Nash equilibrium to transport economics. Bell and Iida call this equilibrium a Deterministic User Equilibrium. The equilibrium assignments in the example are shown as green lines in the figure. On the left, 50 drivers choose each of the two routes, and each driver's trip requires 160 minutes. On the right, all 100 drivers choose the route S, A, B, and D. Each driver takes 205 minutes to complete their trip.

To see why these are equilibria, consider what happens if a single driver deviates from the equilibrium assignment. For example, suppose a driver of the left who has previously chosen the route S, A, and D selects the route S, B, D. The cost for the congested link BD will rise from 50 minutes to 51 minutes, and his total trip time will now be 161 minutes, an increase from the previous 160 minutes. In this model, a driven will not choose to be worse off in this way. Symmetrically, a driver assigned to the route S, B, and D will not decide to switch to the route S, A, and D.

Once the shortcut, AB, has been added, the analysis requires tabulating a few more trips. Suppose a driver swithes from the equilibrium route on the right to the route:

  • S, A, and D or S, B, and D: In each case, the new route includes one congested link which all 100 drivers still traverse. The total trip time is 210 minutes, an undesirable increase over the equilibrium trip time of 205 minutes.
  • S, B, A, and D: All links in this route have a fixed cost. Total trip time is 225 minutes, also an increase over the equilibrium trip time.

So here is a (long-established) case in which improvements to a transport network result in optimizing individuals becoming worse off.

References
  • Satoru Fujishige, Michel X. Goemans, Tobias Harks, Britta Peis, and Rico Zenklusen (2017). Matroids are immune to Braess' Paradox. Mathematics of Operation Research. V. 42, Iss. 3: 745-761.
  • M. G. H. Bell and Y. Iida (1997). Transportation Network Analysis. New York: John Wiley & Sons.

Tuesday, October 24, 2017

Structural Economic Dynamics with a Choice of Technique: A Numerical Example

A Bifurcation Diagram with Two Temporal Paths
I have a working paper with the post title. Here's the abstract:
This article illustrates the application of bifurcation analysis to structural economic dynamics with a choice of technique. A numerical example of the Samuelson-Garegnani model is presented in which technical change is introduced. Examples of temporal paths through the parameter space illustrate variations of the wage frontier. A single technique is initially uniquely cost-minimizing for all feasible rates of profits. Eventually, the technique for which coefficients of production decrease at the fastest rates is always cost-minimizing. During the transition between these positions, reswitching, the recurrence of techniques, and capital-reversing can arise. This example emphasizes the importance of fluke switch points and illustrates possible variations in the existence of Sraffa effects.

Tuesday, October 17, 2017

Elsewhere

  • A July 24 Jonathan Schlefer article, "Market Parables and the Economics of Populism: When Experts are Wrong, People Revolt", in Foreign Affairs. Schlefer cites the Cambridge Capital Controversy as a demonstration that the neoliberal political project of remaking the world around unembedded markets is doomed to failure.
  • A September 11 interview with Daniel Kahneman in which he basically credits Richard Thaler with inventing behavioral economics. (In his memoirs, Misbehaving, Thaler is also explicit about the disciplinary boundaries between economics and psychology.)
  • Richard Thaler's anomaly columns in the Journal of Economic Perspectives
  • I have not read Nancy Maclean's Democracy in Chains. Marshall Steinbaum reviews this book in Boston Review. Henry Farrell & Steven Teles respond.

Another ongoing brouhaha is about Alice and Wu's undergraduate paper documenting the sexism on Economic Job Market Rumors.

Wednesday, October 11, 2017

Others With Points Of View Like Sraffa's

In Production of Commodities by Means of Commodities, Sraffa writes:

"others have from time to time independently taken up points of view which are similar to one or other of those adopted in this paper and have developed them further or in different directions from those proposed here." -- P. Sraffa (1960): pp. vi - vii.

Who is Sraffa talking about? I suggest the following, and their works, at least:

  • Tjalling C. Koopmans (1957). Three Essays on the State of Economic Science. New York: McGraw-Hill
  • Wassily W. Leontief (1941). Structure of the American Economy, 1919-1929.
  • Jacob T. Schwartz (1961). Lectures on the Mathematical Method in Economics. New York: Gordon & Breach.
  • John Von Neumann (1945-1946). A Model of General Economic Equilibrium, A Model of General Economic Equilibrium. V. 13, No. 1: pp. 1-9.

I thought about listing David Hawkins and Herbert Simon, given how frequently the Hawkins-Simon condition is cited in expositions of Leontief input-output analysis. I might also mention Nicholas Georgescu-Roegen, the creator of the non-substitution theorem. The work of Ladislaus Bortkiewicz, Georg von Charasoff, Vladimir K. Dmitriev, and Robert Remak, as I understand it, mostly predates Sraffa's long period of preparation of his masterpiece.

Sunday, October 08, 2017

Economic Impact Of Regional Disasters: A Job For Input-Output Analysis?

This post, unfortunately, is inspired by current events.

Economists can provide guidance on disaster recovery - for example, from earthquakes and hurricanes.

Economists, for a long time, have been developing input-output models of local economies and interactions between them. I think of Walter Isard as a pioneer here. Such models are of practical importance to my post topic.

Regional input-output models can describe disasters with either a supply-side or demand-side approach. In a supply-side approach, the output of an industry is reduced because the inputs into that industry are not available at the pre-disaster level. Some of the outputs of that industry are inputs into other industries. Other outputs satisfy final demands, for example, for household consumption. Input-output modeling can help trace these consequences.

In a demand-side approach, an industry's output is constrained because those who purchase its outputs cannot do so at the pre-disaster level. If those industries who purchase your products are suddenly reduced in size, you must need cut back your output, too.

Some issues arise here. Can supply-side and demand-side approaches be combined without double counting? How should one model the effects of external infusions of aid? Multiplier effects seem to sit comfortably with the demand side approach. The assumption of fixed coefficients in the Leontief input-output approach seems to be an important restriction here. When it comes to modeling resiliency, I think of the work of Adam Rose. Apparently, some use Computational General Equilibrium (CGE) models for this reason.

I do not know enough to have a firm opinion of CGE models. I have the impression that "Computational" is a misnomer; it does not relate to computational theory and Turing machines, as studied in computer science. I am also not sure that the GE in CGE is what I understand as GE. Anyways, practical considerations interact here with the ideological demands impacting economic theory. I like to think that economists are useful in the hard problem of what to do when disaster strikes.

Reference
  • Walter Isard. 1951. Interregional and Regional Input-Output Analysis: A Model of a Space-Economy. Review of Economics and Statistics. Vol. 33, No. 4: pp. 318-328.

Saturday, September 30, 2017

Dean Baker's Rigged And Robert Reich's Saving Capitalism

Dean Baker has a new book out: Rigged: How Globalization and the Rules of the Modern Economy Were Structured to Make the Rich Richer. It recounts how laws that define property, markets, and so on have been rewritten, over the last fifty years, to accomplish an upward redistribution of income. This bias for the rich contrasts with the effects of the rules of the game in the half-century golden age following World War II. This is the same theme as Robert Reich's Saving Capitalism: For the Many, Not the Few. And both books are targeted for the common reader. Thus, a review of Baker's book can usefully compare and contrast it with Reich's book. (I have previously reviewed Reich's book.)

Both books focus on a few areas in which the rules have been rewritten to distribute income upward. For example, consider intellectual property rights, especially the extension of patents and copyrights to last longer and to cover more. Baker, I think, discusses the international dimension more than Reich. The United States has been trying to ensure that patent laws are consistent throughout the world. The largest impact of this attempt, perhaps, is on the price of drugs in developing countries, and the subsequent consequences for health and life.

Both discuss how changes in laws have provided companies with more market power and have protected monopolies and oligopolies. Baker has more of a focus on upper-class professionals, such as doctors, dentists, and lawyers. Baker especially emphasizes how they are protected from international competition. So called free trade treaties, like the North America Free Trade Agreement (NAFTA) are selective in who they subject to the rigors of international competition.

Both discuss corporate governance and the impact on the pay of Corporate Executive Officers (CEOs) and top managers. CEO pay went from 20 times averages wages in the 1960s to over 250 times average wage nowadays. In general, CEO pay is set by a committee appointed by the board of directors who, in turn, are appointed by the CEO. Stockholders have little say, even after the Dodd-Frank bill gives stockholders the right to have an up-or-down non-binding vote on pay packages. Baker extends his critique of CEO pay to heads of foundations and to university presidents, for example.

Reich writes more about contracts, bankruptcy, and enforcement. Baker, on the other hand, writes more about the macroeconomic setting. For example, the Federal Reserve is overly focused on the threat of inflation and not so much on unemployment. I know something of the importance of macroeconomic policy from James Galbraith, who has been writing on this theme for a long time. Baker follows his chapter on macroeconomics with a chapter on the financial sector.

I find Baker more analytical and less polemical than Reich. Baker adopts an interesting trope for putting in context large numbers. He frequently converts dollar flows into multiples of the yearly cost of welfare, that is, the yearly outlay on the Supplemental Nutrition Assistance Program (SNAP). He doesn't always carry through this conversion. I suppose a comparison of doctors' incomes among specialities is only one illustration of health care costs and may not require this contextualization.

I think both books contain a similar tension. Part of their point is that a contrast between non-government intervention in markets and more regulation is a false choice. I think Reich is better on ideological critique of, say, marginal productivity theory or the exploded theory of skills-biased technological change. Baker seems less interested in abstract economic theory, although he does ask whether one can really believe CEOs have gotten so much more productive since the 1950s so as to justify the increased inequality in their pay. But Baker keeps on contrasting legislated barriers to competition with what a free market would produce, that is, less rent. He is too accepting, at least for rhetorical purposes, of traditional economic theory for my tastes. Reich is more consistent with emphasizing that no such thing as a free market can ever exist, absent laws defining markets. Baker does start and conclude with this point.

Baker proposes any number of innovative policies throughout the book, and gathers them together in the second-to-last chapter. For example, Baker suggests that corporations be given an option of issuing non-voting stock to the government, instead of paying corporate income tax. Inventors could be given the option of competing for contest prizes, where a requirement of signing up is that they cannot receive patents over a number of years. (As far as I am aware, existing prize contests have no such connection to the patent system.) He also suggests that governments can pay for medical tourism, where those needing operations travel to other countries in search of cheaper prices. Baker has thought about how some of his policy proposals could first be implemented on a small scale. In general, I find both Baker and Reich too voluntaristic in policy proposals. But I do not know how to avoid that in today's general dismal climate.

I guess my conclusion is that Reich's book is broader, but that Baker's book is generally better in areas of Baker's focus.

Monday, September 18, 2017

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
InputIndustry
IronCorn
AlphaBeta
Labor15,191/5,770305/494
Iron9/201/403/1976
Corn21/10229/494

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
InputIndustry
IronCorn
Labor5/89 ≈ 0.0562 Person-Yrs.57,101/51,353 ≈ 1.11 Person-Yrs.
Iron9/356 ≈ 0.0253 Tons11/356 ≈ 0.0309 Tons
Corn10/89 ≈ 0.112 Bushels11/89 ≈ 0.124 Bushels
Output5/89 ≈ 0.0562 Tons110/89 ≈ 1.24 Bushels

Table 3: Quantity Flows for Beta Technique
InputIndustry
IronCorn
Labor3/577 ≈ 0.00520 Person-Yrs.671/577 ≈ 1.16 Person-Yrs.
Iron27/11,540 ≈ 0.00234 Tons33/11,540 ≈ 0.00286 Tons
Corn6/577 ≈ 0.0104 Bushels2,519/2885 ≈ 0.873 Bushels
Output3/577 ≈ 0.00520 Tons5,434/2,885 ≈ 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
VariableValue
Rate of Profits125,483/209,727 ≈ 59.8 Percent
Wage9,226,807/24,957,513 ≈ 0.370 Bushels per Person-Yr.
Wage7,558/595 ≈ 12.7 Bushels per Ton

Table 5: Aggregates at Switch Point with Real Wicksell Effect of Zero
Technique
AlphaBeta
Net Output1 Bushel Corn
Labor674/577 ≈ 1.17 Person-Years
Physical Capital5/89 Tons Iron3/577 Tons Iron
21/89 Bushels Corn2,549/2,885 Bushels Corn
Financial Capitl113/119 ≈ 0.945 Bushels Corn
Capital-Labor Ratio65,201/80,206 ≈ 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.

Thursday, September 14, 2017

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
InputIndustry
IronCorn
AlphaBeta
Labor1a0,2α305/494
Iron9/201/403/1976
Corna2,11/10229/494

Figure 1 shows two loci in the parameter space defined by the two coefficients of production a0,2α and a2,1. 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 a0,2α 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 a0,2α 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 a0,2α falls below the loci at which the two wage curves are tangent, the wage curves will have two intersections. Suppose a2,1 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 a2,1 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.

Thursday, September 07, 2017

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.

Thursday, August 31, 2017

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
InputIndustry
IronCorn
AlphaBeta
Labor10.802403305/494
Iron9/201/403/1976
Corn3.99737021/10229/494

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
InputIndustries
IronCorn
Labor0.063281.11708
Iron0.028480.03480
Corn0.252960.13922
Outputs0.063281.39217

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
InputIndustries
IronCorn
Labor0.005251.17512
Iron0.002360.00289
Corn0.021000.88230
Outputs0.005251.90330

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
AggregateTechnique
AlphaBeta
Net Output1 Bushel Corn
Labor1.18036 Person-Years
Physical Capital0.06328 Tons
0.39217 Bushels
0.00525 Tons,
0.90330 Bushels
Financial Capital0.94957 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.

Sunday, August 27, 2017

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 a11, β and a11, γ can result in different switch points appearing on the frontier.

Table 1: The Technology for Three of Four Industries
InputIndustry
IronCopperUranium
Labor117,328/8,2811
Iron1/200
Copper0a11, β0
Uranium00a11, γ
Corn000

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
InputProcess
AlphaBetaGamma
Labor1361/913.63505
Iron300
Copper010
Uranium001.95561
Corn000

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, a11, 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, a11, γ, is set to 3/5. This variation in a11, 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 a11, 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 a11, 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 a11, 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 a11, 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 a11, 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 a11, 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 a11, 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 a11, β be set to 51/100, and let the coefficient of production for the input of uranium per unit uranium produced, a11, γ, fall from around 0.55 to 0.4. Figure 7 shows the configuration of wage curves when a11, 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 a11, 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 a11, 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 a11, γ than used in Figure 8, this is a reswitching example between Alpha and Gamma. As a11, 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 a11, β and a11, 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 a11, β and a11, 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
RegionTechniques
1Alpha throughout
2Alpha, Beta, Alpha
3Alpha, Beta
4Beta throughout
5Alpha, Gamma, Alpha
6Alpha, Gamma, Alpha, Beta, Alpha
7Alpha, Gamma, Beta, Alpha
8Alpha, Gamma, Beta
9Alpha, Gamma
10Gamma
11Gamma, 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 a11, 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 a11, 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.