Welcome

I study economics as a hobby. My interests lie in Post Keynesianism, (Old) Institutionalism, and related paradigms. These seem to me to be approaches for understanding actually existing economies.

The emphasis on this blog, however, is mainly critical of neoclassical and mainstream economics. I have been alternating numerical counter-examples with less mathematical posts. In any case, I have been documenting demonstrations of errors in mainstream economics. My chief inspiration here is the Cambridge-Italian economist Piero Sraffa.

In general, this blog is abstract, and I think I steer clear of commenting on practical politics of the day.

I've also started posting recipes for my own purposes. When I just follow a recipe in a cookbook, I'll only post a reminder that I like the recipe.

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.

Piers Anthony, Neoliberal

A Spell for Chameleon, the first book of the Xanth series, shows that Piers Anthony is a neoliberal¹. Magicians are important characters in Xanth, and A Spell introduces us to at least two, Humphrey² and Evil Magician Trent.

We find that "Evil" is just what Trent is called. We are not supposed to regard him as such. And he bases his life entirely on market transactions, even though the setting is a feudal society. Everything is an agreement to a contract, or not, for mutual advantage. An upright person adheres to the spirit of his deals, even when unforeseen circumstances make it unclear what his promises entail in this new situation.

Humphrey is also all about deals. He doesn't like to answer questions, so he always sets the questioner three challenges. Some of these challenges require the questioner to do something for him.

For both Humphrey and Trent, quid pro quo agreements can extend to the most intimate relationships³.

I was prompted to think about neoliberalism by this Mike Konczal article in Vox.

Footnotes

1. One can argue that I am conflating the views of the author with the views of his characters. I think the novels portray both magician Humphrey and Trent in a positive light, but am willing to entertain argument.
2. Humphrey, since he has access to the fountain of youth, as I recall, is an important character throughout the series. I have read hardly any after the first five or ten.
3. Feminists might have something to say about this light reading. The hero, Bink, finds his perfect mate gives him variety, with the young woman's cycle combining certain stereotypical attributes.

Bifurcations Along Wage Frontier

Figure 1: Bifurcation Diagram

1.0 Introduction

This post continues my exploration of the variation in the number and "perversity" of switch points in a model of prices of production. This post presents a case in which one switch point replaces two switch points on the wage frontier.

2.0 Technology

The example in this post is one of an economy in which four commodities can be produced. These commodities are called iron, steel, copper, and corn. The managers of firms know (Table 1) of one process for producing each of the first three commodities. These processes exhibit Constant Returns to Scale. Each column specifies the inputs required to produce a unit output for the corresponding industry. Variations in the parameter d can result in different switch points appearing on the frontier. Each process requires a year to complete and uses up all of its inputs in that year. There is no fixed capital.

Table 1: The Technology for Three of Four Industries

Input	Industry
	Iron	Steel	Copper
Labor	1/2	1	3/2
Iron	d	0	0
Steel	0	1/4	0
Copper	0	0	1/5
Corn	0	0	0

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

Table 2: The Technology the Corn Industry

Input	Process
	Alpha	Beta	Gamma
Labor	1/2	1	3/2
Iron	1/3	0	0
Steel	0	1/4	0
Copper	0	0	1/5
Corn	0	0	0

Four techniques are available for producing a net output of a unit of corn. The Alpha technique consists of the iron-producing process and the Alpha corn-producing process. The Beta technique consists of the steel-producing process and the Beta process. The Gamma technique consists of the remaining two processes.

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

3.0 Result of Technical Progress

Figure 2 shows wage curves when d is 1/3, a fairly high value in this analysis. The wage curves for all three techniques are on the frontier. For certain ranges of the rate of profits, each technique is cost-minimizing. The switch point between the Alpha and Gamma techniques is not on the frontier. No infinitesimal variation in the rate of profits will result in a transition from a position in which the Alpha technique is cost-minimizing in the long period to one in which the Gamma technique is cost-minimizing.

Figure 2: Two Switch Points on Frontier

Suppose technical progress reduces d to 53/180. Figure 3 shows the resulting configuration of the wage curves. There is a single switch point, in which all three wage curves intersect. Aside from the switch point, the Beta technique is no longer cost-minimizing for any other rate of profits.

Figure 3: One Switch Point on Frontier

Figure 4 shows the wage curves when the parameter d has been reduced to 1/5. For d between 53/180 and 1/5, the wage frontier is constructed from the wage curves for the Alpha and Gamma techniques. The Beta technique is never cost minimizing, and the switch point between the Beta and Gamma techniques does not lie on the frontier. The wage curves for the Alpha and Beta techniques have an intersection in the first quadrant only for part of that range for the parameter d. That intersection, however, is never on the frontier for that range. For a value of d less than 1/5, the Alpha technique is dominant. The Beta and Gamma techniques are no longer cost minimizing for any rate of profits.

Figure 4: Bifurcation in which Switch Point on Frontier Disappears

4.0 Conclusion

Figure 1, at the top of the post, summarizes the example. Technical progress can result in a change of the number of switch points, where those switch points disappear and appear along the inside of the wage frontier. Bifurcations need not be across the axes for the wage or the rate of profits.

A Switch Point on the Wage Axis

Figure 1: Bifurcation Diagram

1.0 Introduction

I have been exploring the variation in the number and "perversity" of switch points in a model of prices of production. I conjecture that generic changes in the number of switch points with variations in model parameters can be classified into a few types of bifurcations. (This conjecture needs a more precise statement.) This post fills a lacuna in this conjecture. I give an example of a case that I have not previously illustrated.

2.0 Technology

Consider the technology illustrated in Table 1. The managers of firms know of four processes of production. And these processes exhibit Constant Returns to Scale. The column for the iron industry specifies the inputs needed to produce a ton of iron. In this post, I consider how variations in the parameter e affect the number of switch points. The column for the copper industry likewise specifies the inputs needed to produce a ton of copper. Two processes are known for producing corn, and their coefficients of production are specified in the last two columns in the table. Each process is assumed to require a year to complete and uses up all of its commodity inputs.

Table 1: The Technology for a Three-Commodity Example

Input	Industry
	Iron	Copper	Corn
			Alpha	Beta
Labor	e	3/2	1	3/2
Iron	1/4	0	1/4	0
Copper	0	1/5	0	1/5
Corn	0	0	0	0

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

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

3.0 A Result of Technical Progress

For a high value of the parameter e, the Beta technique minimizes costs, for all feasible wages and rates of profits. Figure 2 illustrates wage curves when e is equal to 21/8. For any wage below the maximum, the Beta technique is cost minimizing. But at a rate of profits of zero, a switch point arises. Both techniques are cost-minimizing.

Figure 2: A Switch Point on the Wage Axis

Suppose technical progress further decreases the person-years needed as input for each ton iron produced. Figure 3 illustrates wage curves when e has fallen to one. For low wages, the Beta technique is cost-minimizing. For high wages, the Alpha technique is preferred. As a result of the structural variation under consideration, the switch point is on the frontier within the first quadrant. It is no longer an intersection of two wage curves with the wage axis.

Figure 3: A Perturbation of the Switch Point on the Wage Axis

By the way, this switch point conforms to outdated neoclassical mumbo jumbo. In a comparison of stationary states, a lower wage around the switch point is associated with the adoption of a more labor-intensive technique. When analyzing switch points, this is a special case with no claim to logical necessity. John Cochrane and Bryan Caplan are ignorant of price theory. Contrast with Steve Fleetwood.

4.0 Conclusion

Technical progress can result in a new switch point appearing over the axis for the wage. Given a stationary state, this switch point is "non-perverse" until the occurrence of another structural bifurcation.

Generic Bifurcations and Switch Points

This post states a mathematical conjecture.

Consider a model of prices of production in which a choice of technique exists. The parameters of model consist of coefficients of production for each technique and given ratios for the rates of profits among industries. The choice of technique can be analyzed based on wage curves. A point that lies simultaneously on the outer envelope of all wage curves and the wage curves for two techniques (for non-negative wages and rates of profits not exceeding the maximum rates of profits for both techniques) is a switch point.

Conjecture: The number of switch points is a function of the parameters of the model. The number of switch points varies with variations in the parameters.

• A pair of switch points can arise if:
• One wage curve dominates another for one set of parameter values.
• The wage curves become tangent at a single switch point, for a change in one parameter.
• The point of tangency breaks up into two switch points (reswitching) as that parameter continues in the same direction.
• A switch point can disappear (for an economically relevant ranges of wages) if:
• A switch point exists for some set of parameter values.
• For some variation of a parameter, that switch point becomes the intersection of both wage curves with one of the axes (the wage or the rate of profits).
• A further variation of the parameter in the same direction leads to the point of intersection of the wage curves falling out of the first quadrant.
• Like the above, but a switch point can disappear if a variation in a parameter results in that intersection of two wage curves falling off the outer envelope. (A third wage curve becomes dominant for the wage at which the intersection occurs.)

The above three possibilities are the only generic bifurcations in which the number of switch points can change with model parameters.

Proof: By incredibility. How could it be otherwise?

I claim that the above conjecture applies to a model with n commodities, not just the two-commodity example I have previously analyzed. It applies to a choice among as many finite techniques as you please. Different techniques may require different capital goods as inputs. Not all commodities need be basic.

In actuality, I do not know how to prove this. I am not sure what it means for a bifurcation to be generic in the above conjecture, but I want to allow for a combination of, say, two of the three possibilities. For example, the point of tangency for two wage curves (in the first case) may simultaneously be the intersection of both wage curves with the axis for the rate of profits. In this case, only one switch point arises with continuous variation of model parameters; the other falls below the axis for the rate of profits. I want to say such a bifurcation is non-generic, in some sense.

This post needs pictures. I assume the third possibility can arise for some parameter in at least one of these examples. (Maybe I need to think harder to be sure that the number of switch points changes. What do I want to say is non-generic here?) I have an example in which a switch point disappears by falling below the axis for the rate of profits, but I do not have an example of a switch point disappearing by crossing the wage axis.

Voting Efficiency Gap: A Performative Theory?

Table 1: Distribution of Votes Among Parties and Districts

District	Tories	Whigs	Total
I	51	49	100
II	51	49	100
III	33	67	100
Total	135	165	300

1.0 Introduction

This post, amazingly enough, is on current events. Stephanopoulos and McGhee have developed a formula, the efficiency gap, that measures the partisanship of the lines drawn for legislative districts. In this post, I present a numerical illustration of this formula and connect it to current events. I conclude with some questions.

2.0 Numerical Example

Consider a population of 300 voters divided between two parties. The Whigs are in the majority, with 55% of the electorate. Suppose the government has a three-member council, with each member elected from a district. And each district contains 100 voters.

2.1 Drawing Districts

The Tories, despite being the minority party have drawn the districts. The votes in the last election are as in Table 1. The Tories are in the minority of the population, but hold two out of three council seats.

The Tories, in this example, cannot win all seats. In the seats they lose, they want to pack as many Whigs as possible. So where the Whigs win, they win overdominatingly. Many of the Whig votes in that single district are wasted on running up a victory more than necessary. On the other hand, the Tories try to draw their winning districts to win as narrowly as possible. The Whig votes in the districts in which the Whigs lose are said to be cracked.

This is an extreme example, sensitive to small variations in the districts in which the Tories win. They would probably want safer majorities in those districts.

As far as I can see, the drawing of odd-shaped district lines is not necessary for gerrymandering. Consider a city surrounded by suburbs and a rural area. Suppose, that downtown tends to vote differently than the suburbs and rural areas. One could imagine district lines drawn outward from the central city. Depending on relative populations, that might distribute the urban voters such that they predominate in all districts. On the other hand, one might create a few compact districts in the center to pack many urban voters, with the ones remaining in cropped pizza slices having their votes cracked.

2.2 Wasted Votes

Define a vote to be wasted if either it is for a losing candidate in your district or it is for a winning candidate, but it exceeds the number needed for a majority in that district. The number of wasted votes for each party in the numerical example is:

• The Tories have 33 wasted votes.
• The Whigs have 49 + 49 + (67 - 51) = 114 wasted votes.

The efficiency gap is a single number that combines the number of wasted votes in both parties. An invariance property arises here. As I have defined it, the number of wasted votes, summed across parties, in each district is 49. Forty nine is one less than half the number of votes in a district. This is no accident.

2.3 Arithmetic

In calculating the efficiency gap, one takes the absolute value of the difference between the parties in the number of wasted votes. In the example, this number is | 33 - 114 | = 81.

The efficiency gap is the ratio of this positive difference to the number of voters. So the efficiency gap in the example is 81/300 = 27%.

3.0 Contemporary Relevance in the United States

The United States Supreme Court has decided, in a number of cases over the last decades, that gerrymandering might be something they can rule on. Partisan redistricting is not purely a political issue that they do not want to get involved in. Apparently, however, they have never found a clear example.

But what is gerrymandering? Can they define some sort of rule that lower courts can use? How would politicians drawing up district lines know whether or not their decisions will withstand challenges in court? Apparently, Justice Kennedy, among others expressed a hankering for some such rule in his decision in League of United Latin American Citizens (LULAC) vs. Perry (2006).

Gill vs. Whitford is a current case on the Supreme Court docket. And the efficiency gap, which is relatively new mathematics, may be discussed in the pleadings, at least, in this case.

So the creation of the mathematical formula illustrated above might affect the law in the United States. If so, it will impact how districts are drawn and what some consider fair. It is interesting that I can now raise the issue of the performativity of mathematics in a non-historical context, while the mathematics is, perhaps, performing.

4.0 Questions

I am working on reading two of the three references below. (Articles in law reviews seem to be consistently lengthy.) I have some questions and comments.

Berstein and Duchin (2017) seems to raise some severe objections. Suppose the election in a district with 100 voters is decided either 75 to 25 or 76 to 24. The way I have defined it, the difference in wasted votes in this district is (24 - 25) or (25 - 24). That is, this district contributes one vote to the difference in wasted votes. So the definition of the efficiency gap privileges races that are won with 75% of the vote.

Consider a case in which one party has support from 75 percent of the voters. Suppose the districts are drawn such that each district casts 75% of their votes for that party. So this party wins 100% of the seats and the efficiency gap is minimized. Do we want to say this is not an example of gerrymandering?

Is the efficiency gap related to power indices somehow or other? How should the efficiency gap be calculated if more than two parties are contesting an election? Mayhaps, one should calculate the efficiency gap for each pair of parties. This loses the simplicity of a single number. Also, sometimes clever Republican strategists might try to help themselves by helping the Green Party, at the expense of the Democratic Party. How does this measure compare and contrast with other measures? As I understand it, a measure of partisan swing, for example, relies on counterfactuals, while the efficiency gap is not counterfactual.

References

• Mira Bernstein and Moon Duchin (2017). A Formula Goes to Court: Partisan Gerrymandering and the Efficiency Gap.
• Geometry of Redistricting, Tufts University, to be held 7-11 August 2017
• Nicholas Stephanopoulos and Eric McGhee (2015). Partisan Gerrymandering and the Efficiency Gap, University of Chicago Law Review.

Bifurcations And Switchpoints

I have organized a series of my posts together into a working paper, titled Bifurcations and Switch Points. Here is the abstract:

This article analyzes structural instabilities, in a model of prices of production, associated with variations in coefficients of production, in industrial organization, and in the steady-state rate of growth. Numerical examples are provided, with illustrations, demonstrating that technological improvements or the creation of differential rates of profits can create a reswitching example. Variations in the rate of growth can change a "perverse" switch point into a normal one or vice versa. These results seem to have implications for the stability of short-period dynamics and suggest an approach to sensitivity analysis for certain empirical results regarding the presence of Sraffa effects.

Here are links to previous expositions of parts of this analysis:

• Variation in selected coefficients of production:
• Variation of switch points
• A bifurcation diagram
• A model of prices of production under oligopoly:
• The model
• Variation of switch points with markups
• A bifurcation diagram
• Bifurcations with variations in the rate of growth

In comments, Sturai suggests additional research with the model of oligopoly. One could take the standard commodity as such that it has no markup. What I am calling the scale factor for the rate of profits would be the rate of profits made in the production of the standard commodity. Markups for individual industries would be based on this. I have identified a problem, much like the transformation problem, in comparing and contrasting free competition and oligopoly. I would have to think about this.