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.
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 |
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 TechnologyThe 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.
Input | Corn Industry | ||
Alpha | Beta | Gamma | |
Labor | 1 | 3.69174 | 3.33574 |
Iron | 3 | 0 | 1 |
Copper | 0 | 0.92850 | 0 |
Uranium | 0 | 0 | 1.79455 |
Corn | 0 | 0 | 0 |
Input | Industry | ||
Iron | Copper | Uranium | |
Labor | 1 | 1.94290 | 0.917647 |
Iron | 0.5 | 0 | 1 |
Copper | 0 | 0.5 | 0 |
Uranium | 0 | 0 | 0.550588 |
Corn | 0 | 0 | 0 |
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 FrontierFor 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 X_{SA} and X_{SA} 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.
Link | Cost |
SA | X_{SA} Minutes |
SB | 110 Minutes |
AD | 110 Minutes |
BD | X_{BD} Minutes |
AB | Either 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 |
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.