Wednesday, January 01, 2020
Welcome
The emphasis on this blog, however, is mainly critical of neoclassical and mainstream economics. I have been alternating numerical counter-examples with less mathematical posts. In any case, I have been documenting demonstrations of errors in mainstream economics. My chief inspiration here is the Cambridge-Italian economist Piero Sraffa.
In general, this blog is abstract, and I think I steer clear of commenting on practical politics of the day.
I've also started posting recipes for my own purposes. When I just follow a recipe in a cookbook, I'll only post a reminder that I like the recipe.
Comments Policy: I'm quite lax on enforcing any comments policy. I prefer those who post as anonymous (that is, without logging in) to sign their posts at least with a pseudonym. This will make conversations easier to conduct.
Tuesday, December 12, 2017
An Example of Bifurcation Analysis with Land and the Choice of Technique
Figure 1: A Bifurcation Diagram |
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 ModelTable 1 specifies the technology for this example. One parameter, the labor coefficient a_{0}^{β}, 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.
Input | Corn Industry | ||
Alpha | Beta | ||
Labor | a_{0}^{α} = 1 Person-Yr. | a_{0}^{β} Person-Yr. | |
Land | b_{α} = 10 Acres of Type I | b_{β} = 20 Acres of Type II | |
Corn | a_{α} = (1/4) Bushels | a_{β} = (1/5) Bushels |
Each type of land is in fixed supply:
- L_{I} = 100 Acres of Type I land exist.
- L_{II} = 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 - a_{α}) L_{I}/b_{α} = (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 a_{0}^{β} may fall, I am assuming increased productivity (per worker) is taken in the form of decreased employment.
3.0 Price EquationsI 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 ρ_{I} + w = 1
(1/5)(1 + r) + 20 ρ_{II} + a_{0}^{β} w = 1
where r, w, ρ_{I}, and ρ_{II} 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:
ρ_{I} ρ_{II} = 0
4.0 The Choice of Technique
I consider three solutions of the price equation, each for a different parameter value of a_{0}^{β}.
4.1 First ExampleFirst, suppose a_{0}^{β} 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 AxisI next consider a parameter value for a_{0}^{β} 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 a_{0}^{β} 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"
r_{switch} = (15 a_{0}^{β} - 16)/(5 a_{0}^{β} - 4)
This function asymptotically approaches the maximum rate of profits for the Alpha technique as a_{0}^{β} 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:
ρ_{I} = [(15 a_{0}^{β} - 16) + (4 - 5 a_{0}^{β})r]/(200 5 a_{0}^{β})
When the rent on Type II land is non-negative, it is:
ρ_{II} = [(16 - 15 a_{0}^{β}) + (5 a_{0}^{β} - 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.
Region | Cost-Minimizing Techniques | Notes |
1 | Alpha | One technique cost-minimizing. |
2 | Alpha, Beta | "Normal" switch point. |
3 | Beta, Alpha, Beta | Reswitching. Switch pt. at highest r is "perverse". |
4 | Beta | One technique cost-minimizing. |
5 | Alpha, Gamma | "Normal" switch point. |
6 | Alpha, Gamma, Beta | "Normal" switch points. |
7 | Beta, Alpha, Gamma, Beta | Recurrence of techniques. Switch pt. at highest r is "perverse". |
8 | Beta, Alpha, Beta, Gamma, Beta | Two reswitchings, two "perverse" switch pts. |
9 | Gamma | One technique cost-minimizing. |
10 | Gamma, Beta | "Normal" switch point. |
11 | Beta, Gamma, Beta | Reswitching. 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.
Input | Iron Industry | Corn Industry | ||
Alpha | Beta | Gamma | ||
Labor | 1 | 0.89965 | 0.71733 e^{-σ t} | 1.28237 e^{-φ t} |
Iron | 0.45 | 0.025 | 0.00176 e^{-σ t} | 0.03375 e^{-φ t} |
Corn | 2 | 0.1 | 0.53858 e^{-σ t} | 0.13499 e^{-φ t} |
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 |
Region | Cost-Minimizing Techniques | Notes |
1 | Alpha | One technique cost-minimizing. |
2 | Alpha, Beta | "Normal" switch point. |
3 | Beta, Alpha, Beta | Reswitching. Switch pt. at highest r is "perverse". |
4 | Beta, Alpha, Gamma, Beta | Recurrence of techniques. Switch pt. at highest r is "perverse". |
5 | Beta, Gamma, Beta | Reswitching. Switch pt. at highest r is "perverse". |
6 | Gamma, Beta | "Normal" switch point. |
7 | Gamma | One 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.