Wednesday, January 01, 2020

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.

Wednesday, July 26, 2017

The Choice Of Technique With Multiple And Complex Interest Rates

My article with the post title is now available on the website for the Review of Political Economy. It will be, I gather, in the October 2017 hardcopy issue. The abstract follows.

Abstract: This article clarifies the relations between internal rates of return (IRR), net present value (NPV), and the analysis of the choice of technique in models of production analyzed during the Cambridge capital controversy. Multiple and possibly complex roots of polynomial equations defining the IRR are considered. An algorithm, using these multiple roots to calculate the NPV, justifies the traditional analysis of reswitching.

Sunday, July 23, 2017

A Switch Point Disappearing Over The Wage Axis

Figure 1: Bifurcation Diagram
1.0 Introduction

In a series of posts, I have been exploring structural economic dynamics. Innovation reduces coefficients of production. Such reductions can vary the number and sequence of switch points on the wage frontier. I call such a variation a bifurcation. And I think such bifurcations, at least if only one coefficient decreases, fall into a small number of normal forms.

One possibility is that a decrease in a coefficient of production results in a switch point appearing over the wage axis, as illustrated here. This post modifies that example such that the switch point disappears over the wage axis with a decrease in a coefficient of production.

2.0 Technology

Accordingly, consider the technology illustrated in Table 1. The managers of firms know of four processes of production. And these processes exhibit Constant Returns to Scale. The column for the iron industry specifies the inputs needed to produce a ton of iron. The column for the copper industry likewise specifies the inputs needed to produce a ton of copper. In this post, I consider how variations in the parameter u affect the number of switch points. 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
InputIndustry
IronCopperCorn
AlphaBeta
Labor21/8u13/2
Iron1/401/40
Copper01/501/5
Corn0000

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 Results

Consider variations in u, the input of labor in the copper industry, per unit copper produced. Figure 1 shows the effects of such variations. For a high value of this coefficient, a single switch point exists. The Alpha technique is cost-minimizing at high wages (or low rates of profits). The Beta technique is cost-minimizing at low wages (or high rates of profits).

Suppose that technical innovations reduce u to 3/2. Then the switch point occurs at the maximum wage. For all positive rates of profits (not exceeding the maximum), the Beta technique is cost-minimizing. At a rate of profits of zero, both techniques (or any linear combination of them) are eligible for adoption by cost-minimizing firms.

A third regime arises when technical innovations reduce u even more. The a technique is cost-minimizing for all feasible rates of profits, including a rate of profits of zero.

4.0 Discussion

So this example has illustrated that the bifurcation diagram at the top of this previous post can be reflected across a vertical line where the bifurcation occurs. An abstract description of a bifurcation in which a switch point crosses the wage axis does not have a direction, in some sense. Either direction is possible.

The illustrated bifurcation is, in some sense, local. The illustrated phenomenon might occur in what is originally a reswitching example. That is, the bifurcation concerns only what happens around a small rate of profit (or near the maximum wage). It is compatible with wage curves that have a second intersection on the frontier at a higher rate of profits. In such a case, the switch point at the higher rate of profits will remain. But the bifurcation will transform it from a 'perverse' switch point to a 'normal' one.

As I understand it, such a bifurcation of a reswitching will be manifested in the labor market with 'paradoxical' behavior. Suppose the first switch point disappears over the wage axis. Around the second switch point, a comparison of long period (stationary) positions will find a higher wage associated with the adoption of a technique that requires less labor per (net) unit output, for the economy as a whole. But, in the corn industry, a higher wage will be associated with the adoption of a technique that requires more labor per (gross) unit corn produced.

This is just one of those possibilities that demonstrates the Cambridge Capital Controversy is not merely a critique of aggregation, macroeconomics, and the aggregate production function. It has implications for microeconomics, too.

Thursday, July 20, 2017

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.

Sunday, July 16, 2017

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
InputIndustry
IronSteelCopper
Labor1/213/2
Irond00
Steel01/40
Copper001/5
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
Labor1/213/2
Iron1/300
Steel01/40
Copper001/5
Corn000

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.

Tuesday, July 11, 2017

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
InputIndustry
IronCopperCorn
AlphaBeta
Labore3/213/2
Iron1/401/40
Copper01/501/5
Corn0000

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.

Saturday, July 08, 2017

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.