Tuesday, August 15, 2017

Elsewhere

  • Nick Hanauer argues for some policies that postulate:
    • Income distribution is not a matter of supply and demand or any other sort of economic natural laws.
    • That a more egalitarian distribution of income leads to an increased demand and generalized shared prosperity.
  • Tom Palley contrasts neoliberalism with an economic theory with an approach with another "theory of income distribution and its theory of aggregate employment determination".
  • Elizabeth Bruenig contrasts liberalism with the the left.
  • Paul Blest laughs at whining neoliberals
  • Chris Lehmann considers how the turn of the US's Democratic Party to neoliberalism lowers its electoral prospects.

Is the distinction between democratic socialism and social democracy of no practical importance at the moment in any nation's politics? I think of the difference in two ways. First, in the United States in the 1970s, leftists had an argument. Self-defined social democrats became Neoconservatives, while democratic socialists found the Democratic Socialists of America (DSA). Second, both are reformists approaches to capitalism, advocating tweaks to, as Karl Popper argued for, prevent unnecessary pain. But social democrats have no ultimate goal of replacing capitalism, while democratic socialists want to end up with a transformed system.

Saturday, August 12, 2017

A Fluke Of A Fluke Switch Point

Figure 1: Wage Curves
1.0 Introduction

This post presents an example of the analysis of the choice of technique in competitive markets. The example is one with three techniques and two switch points. The wage curves for the Alpha and Beta techniques are tangent at one of the switch points. This is a fluke. And the wage curves for all three techniques all pass through that same switch point. This, too, is a fluke.

I suppose that the example is one of reswitching and capital-reversing is the least interesting property of the example. Paul Samuelson was simply wrong in labeling such phenomena as perverse. A non-generic bifurcation, like the illustrated one, falls out of a comprehensive analysis of possible configurations of wage curves.

2.0 Technology

The technology in the example has a particularly simple structure. Firms can produce one of three capital goods, which I am arbitrarily labeling iron, copper, and uranium. Table 1 shows the production processes known for producing each metal. One process is known for producing each, and each metal is produced out of inputs of labor and that metal. Each process requires a year to complete, uses up all its material inputs, and exhibits Constant Returns to Scale.

Table 1: The Technology for Three of Four Industries
InputIndustry
IronCopperUranium
Labor117,328/8,2811
Iron1/200
Copper048/910
Uranium000.53939
Corn000

Three processes are known for producing corn (Table 2), which is the consumption good. This economy can be sustained by adopting one of three techniques. 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. Finally, the Gamma technique consists of the remaining two processes.

Table 2: The Technology the Corn Industry
InputProcess
AlphaBetaGamma
Labor1361/913.63505
Iron300
Copper010
Uranium001.95561
Corn000
3.0 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-rate of profits for the three 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. Aside from switch points, the Alpha technique is cost-minimizing at low and high rates of profits, with the Gamma technique cost-minimizing between the switch points. At switch points, any linear combination of the techniques with wage curves going through that switch point are cost-minimizing.

The wage curve for the Beta technique is a straight line. This affine property results from the Organic Composition of Capital being the same in copper production and in corn production, when the Beta technique is adopted. To help visualization, I also graph the difference between the wage curves (Figure 2). The Beta technique is only cost-minimizing at the switch point at the higher rate of profits. The tangency of the wage curves for the Alpha and Beta techniques is manifested in Figure 2 by the non-negativity of the difference in these curves.

Figure 2: Distance Between Wage Curves

4. Conclusion

I'm sort of proud of this example. I suppose I could, at least, submit it for publication somewhere. But it is only a side effect of a larger project I guess I am pursuing.

I want to introduce a distinction among fluke switch points. Every bifurcation (that is, a change in the sequence of switch points and cost-minimizing techniques along the wage frontier) is a fluke. Some perturbation of a coefficient of production from a bifurcation value will change that sequence. Suppose a perturbation of a coefficient of production not involved in a bifurcation, in some sense, leaves the qualitative story unchanged. One can use the same bifurcation to tell a story about, say, technological progress. This is a generic bifurcation.

Accept, for the sake of argument, that prices of production tell us something about actual prices. The economy is never in an equilibrium, but owners of firms are always interested in increasing their profits. One can never expect observed technology to meet the fluke conditions of a generic bifurcation. But it can tell us something about how the dynamics of income distribution, for example, vary with technological progress.

Suppose one perturbs, in the example, the coefficient of production for the amount of iron needed to produce iron. (I denote this coefficient, in a fairly standard notation, as a1,1β.) Then, either the wage curves for the. Alpha and Beta techniques will not intersect at all or they will intersect twice. In the latter case, one can vary a1,1γ to find an example in which all three wage curves intersect at one or another of the switch points. But the tangency will be lost. So I consider the fluke point illustrated to be a non-generic bifurcation.

Non-generic bifurcations arise in a complete bifurcation analysis. The model illustrated remains open. Income distribution is not specified. Nevertheless, I think this theoretical analysis can say something to those who are attempting to empirically apply the Leontief-Sraffa model.

Monday, August 07, 2017

Some Unresolved Issues In Multiple Interest Rate Analysis

1.0 Introduction

Come October, as I understand it, the Review of Political Economy will publish, in hardcopy, my article The Choice of Technique with Multiple and Complex Interest Rates. I discuss in this post questions I do not understand.

2.0 Non-Standard Investments and Fixed Capital

Consider a point-input, flow-output model. In the first year, unassisted labor produces a long-lived machine. In successive years, labor and a machine of a specific history are used to produce outputs of a consumption good and a one-year older machine. The efficiency of the machine may vary over the course of its physical lifetime. When the machine should be junked is a choice variable in some economic models.

I am aware that in this, or closely related models, the price of a machine of a specific date can be negative. The total value of outputs at such a year in which the price of the machine of the machine is negative, however, is the difference between the sum of the price of the machine & the consumption good and the price of any inputs, like labor, that are hired in that year. Can one create a numerical example of such a case in which the net value, in a given year, is negative, where that negative value is preceded and followed by years with a positive net value?

If so, this would an example of a non-standard investment. A standard investment is one in which all negative cash flows precede all positive cash flows. In a non-standard investment at least one positive cash flow precedes a negative cash flow, and vice-versa. Non-standard investments create the possibility that all roots of the polynomial used to define the Internal Rate of Return (IRR) are complex. Can one create an example with fixed capital or, more generally, joint production in which this possibility arises?

Does corporate finance theory reach the same conclusions about the economic life of a machine as Sraffian analysis in such a case? Can one express Net Present Value (NPV) as a function combining the difference between the interest rate and each IRR in this case, even though all IRRs are complex? (I call such a function an Osborne expression for the NPV.)

3.0 Generalizing the Composite Interest Rate to the Production of Commodities by Means of Commodities

In my article, I follow Michael Osborne in deriving what he calls a composite interest rate, that combines all roots of the polynomial defining the IRR. I disagree with him, in that I do not think this composite interest rate is useful in analyzing the choice of technique. But we both obtain, in a flow-input, point output model, an equation I find interesting.

This equation states that the difference between the labor commanded by a commodity and the labor embodied in that commodity is the product of the first input of labor per unit output and the composite interest rate. Can you give an intuitive, theoretical explanation of this result? (I am aware that Osborne and Davidson give an explanation, that I can sort of understand when concentrating, in terms of the Austrian average period of production.)

A model of the production of commodities by means of commodities can be approximated by a model of a finite sequence of labor inputs. The model becomes exact as the number of dated labor inputs increases without bound. In the limit, the labor command by each commodity is a finite value. So is the labor embodied. And the quantity for the first labor input decrease to zero. Thus, the composite interest rate increases without bound. How, then, can the concept of the composite interest rate be extended to a model of the production of commodities by means of commodities?

4.0 Further Comments on Multiple Interest Rates with the Production of Commodities by Means of Commodities

In models of the production of commodities by means of commodities, various polynomials arise in which one root is the rate of profits. I have considered, for example, the characteristic equation for a certain matrix related to real wages, labor inputs, and the Leontief input-output matrix associated with a technique of production. Are all roots of such polynomials useful for some analysis? How so?

Luigi Pasinetti, in the context of a theory of Structural Economic Dynamics, has described what he calls the natural system. In the price system associated with the natural system, multiple interest rates arise, one for each produced commodity. Can these multiple interest rates be connected to Osborne's natural multiple interest rates?

5.0 Conclusion

I would not mind reading attempts to answer the above questions.

Friday, August 04, 2017

Switch Points and Normal Forms for Bifurcations

I have put up a working paper, with the post title, on my Social Sciences Research Network (SSRN) site.

Abstract: The choice of technique can be analyzed, in a circulating capital model of prices of production, by constructing the wage frontier. Switch points arise when more than one technique is cost-minimizing for a specified rate of profits. This article defines four normal forms for structural bifurcations, in which the number and sequence of switch points varies with a variation in one model parameter, such as a coefficient of production. The 'perversity' of switch points that appear on and disappear from the wage frontier is analyzed. The conjecture is made that no other normal forms exist of codimension one.

Tuesday, August 01, 2017

Switch Points Disappearing Or Appearing Over The Axis For The Rate Of Profits

Figure 1: Two Bifurcation Diagrams Horizontally Reflecting
1.0 Introduction

This post continues my investigation of structural economic dynamics. I am interested in how technological progress can change the analysis of the choice of technique. In this case, I explore how a decrease in a coefficient of production can cause a switch point to appear or disappear over the axis for the rate of profits.

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. 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. Technology is defined in terms of two parameters, u and v. u denotes the quantity of iron needed to produce a unit iron in the iron industry. v is the quantity of copper needed to produce a unit copper.

Table 1: The Technology for a Three-Commodity Example
InputIndustry
IronCopperCorn
AlphaBeta
Labor12/312/3
Ironu01/30
Copper0v01/3
Corn0000

As usual, 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.

I make all my standard assumptions. 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 Innovations

I have two stories of technical innovation. In one, improvements are made in the process for producing copper. As a consequence, the wage curve for the Beta technique moves outward. In the other story, improvements are made in the iron industry, and the wage curve for the Alpha technique moves outwards. The bifurcations that occur in the two stories are mirror reflections of one another, in some sense.

3.1 Improvements in Copper Production

Let u be fixed at 1/3 tons per ton. The wage curve for the Alpha technique is a downward sloping straight line. Let v decrease from 1/2 to 3/10. When v is 1/3, the wage curve for the Beta technique is also a straight line. I created the example to have linear (actually, affine) wage curves at the bifurcation for convenience. The bifurcation does not require such.

Figure 2 shows the wage curves when the copper coefficient for copper production is a high value, in the range under consideration. A single switch point exists, and the Alpha technique is cost-minimizing if the rate of profits is high. As v decreases, the switch point moves to a higher and higher rate of profits. (These statements are about the shapes of mathematical functions. They are not about historical processes set in time.) Figure 3 shows the wage curves when v is 1/3. The switch point is now on the axis for the rate of profits. For any non-negative rate of profits below the maximum, the Beta technique is cost-minimizing. Finally, Figure 4 shows the wage curves for an even lower copper coefficient in copper production. Now, there is no switch point, and the Beta technique is always cost-minimizing, for all possible prices of production.

Figure 2: Wage Curves Without Improvement in Copper Production

Figure 3: Wage Curves For A Bifurcation

Figure 4: Wage Curves After Improvements in Copper Production

3.2 Improvements in Iron Production

Now let v be set at 1/3. Let u decrease from 1/2 to 3/10. Figure 5 shows the wage curves at the high end for the iron coefficient in iron production. No switch point exists, and the Beta technique is always cost-minimizing. I thought about repeating Figure 3, for v decreased to 1/3. The same configuration of wage curves, with a bifurcation, appears in this story. Figure 6, shows that the switch point appears for an even lower value of the iron coefficient.

Figure 5: Wage Curves Without Improvement in Iron Production

Figure 6: Wage Curves After Improvements in Iron Production

3.3 Improvements in Both Iron and Copper Industries

I might as well graph (Figure 7) the copper coefficient in copper production against the iron coefficient in iron production. The bifurcation occurs when the maximum rates of profits are identical in the Alpha and Beta technique. In a model with the simple structure of the example, this occurs when u = v. Representative illustrations of wage curves are shown in the regions in the parameter space. A switch point below the maximum rate of profits exists only above the line in parameter space representing the bifurcation.

Figure 7: Bifurcation Diagram for Two Coefficients of Production

The story in Section 3.1 corresponds to moving downwards on a vertical line in Figure 7. The left-hand side of Figure 1, at the top of this post, is another way of illustrating this story. On the other hand, Section 3.2 tells a story of moving leftwards on a horizontal line in Figure 7. The right-hand side of Figure 1 illustrates this story.

Focus on the intersections, in the two sides of Figure 1 of the blue, red, and purple loci. Can you see that, in some sense, they are reflections, up to a topological equivalence?

4.0 Discussion

I have a reswitching example with a switch point disappearing over the axis for the rate of profits. In that example, the disappearing switch point is 'perverse', that is, it has a positive real Wicksell effect. In the examples in Section 3 above, the disappearing or appearing switch point is 'normal', with a negative real Wicksell effect.

Friday, July 28, 2017

Bifurcations Along Wage Frontier, Reflected

Figure 1: Bifurcation Diagram
1.0 Introduction

This post continues a series investigating structural economic dynamics. I think most of those who understand prices of production - say, after working through Kurz and Salvadori (1995) - understand that technical innovation can change the appearance of the wage frontier. (The wage frontier is also called the wage-rate of profits frontier and the factor-price frontier.) Changes in coefficients of production can create or destroy a reswitching example. But, as far as I know, nobody has systematically explored how this happens in theory.

I claim that when switch points appear on or disappear off of the wage frontier, these bifurcations follow a few normal forms. I have been describing each normal form as a story of a coefficient of production being reduced by technical innovation. I further claim that, in some sense, the order of changes along the wage frontier is not specified. One can find an example with a decreasing coefficient of production in which the order is the opposite of some other example of technical innovation with the same normal form.

This post is one of a series providing the proof that order does not matter. The example in this post relates to this previous example.

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 v 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
Iron53/18000
Steel0v0
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 Technical Progress

Figures 2 through 5 illustrate wage curves for different levels of the coefficient of production denoted v in the table. Figure 2 shows that for a relatively high parameter value, the switch point between the Alpha and Gamma techniques is the only switch point on the outer frontier. For continuously lower parameter values of v, the wage curve moves outward. Figure 3 illustration the bifurcation value, a fluke case in which the wage curves for all three techniques intersect in a single switch point. Other than at the switch point, the wage curve for the Beta technique is not on the frontier. But, for a slightly lower parameter value (Figure 4), the wage curve for the Beta technique, along with switch points between the Alpha and the Beta techniques and between the Beta and Gamma techniques, is on the frontier. The intersection between the wage curves for the Alpha and Gamma techniques is no longer on the frontier. Figure 5 illustrates another bifurcation in the example. The focus of this post is not on this bifurcation, in which a switch point disappears over the axis for the rate of profits.

Figure 2: Wage Curves with High Steel Inputs in Steel Production

Figure 3: Wage Curves with Medium Steel Inputs in Steel Production

Figure 4: Wage Curves with Low Steel Inputs in Steel Production

Figure 5: Wage Curves with Lowest Steel Inputs in Steel Production

4.0 Discussion

The bifurcation on the right in Figure 1, at the top of this post, is topologically equivalent to the horizontal reflection of the bifurcation on the right in the equivalent figure in this previous post. (On the other hand, the bifurcations on the upper left in both diagrams are the same normal form, in the same order.)

The bifurcation described in this post is a local bifurcation. To characterize this bifurcation, one need only look at small range of rates of profits and coefficients of production around a critical value. Accordingly, then wage curves involved in the bifurcations could intercept any number of times, in some other example of this normal form, at positive rates of profits. Each of the three switch points involved in the bifurcation could have any direction for real Wicksell effects, positive or negative.

The bifurcation, as depicted in this post, replaces one switch point on the wage curve with two switch points. It could be that the switch point disappearing exhibits capital-reversing, and both of the two new switch points appearing also exhibit capital-reversing. But any of five other other combinations are possible.

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.

Tuesday, July 04, 2017

Voting Efficiency Gap: A Performative Theory?

Table 1: Distribution of Votes Among Parties and Districts
DistrictToriesWhigsTotal
I5149100
II5149100
III3367100
Total135165300
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

Friday, June 30, 2017

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:

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.

Saturday, June 24, 2017

Bifurcation Analysis in a Model of Oligopoly

Figure 1: Bifurcation Diagram

I have presented a model of prices of production in which the the rate of profits differs among industries. Such persistent differential rates of profits may be maintained because of perceptions by investors of different levels of risk among industries. Or they may reflect the ability of firms to maintain barriers to entry in different industries. In the latter case, the model is one of oligopoly.

This post is based on a specific numeric example for technology, namely, this one, in which labor and two commodities are used in the production of the same commodities. I am not going to reset out the model here. But I want to be able to refer to some notation. Managers know of two processes for producing iron and one process for producing corn. Each process is specified by three coefficients of production. Hence, nine parameters specify the technology, and there is a choice between two techniques. In the model:

  • The rate of profits in the iron industry is rs1.
  • The rate of profits in the corn industry is rs2.

I call r the scale factor for the rates of profits. s1 is the markup for the rate rate of profits in the iron industry. And s2 is the markup for the rate of profits in the corn industry. So, with the two markups for the rates of profits, 11 parameters specify the model.

I suppose one could look at work by Edith Penrose, Michal Kalecki, Joseh Steindl, Paolo Sylos Labini, Alfred Eichner, or Robin Marris for a more concrete understanding of markups.

Anyways, a wage curve is associated with each technique. And that wage curve results in the wage being specified, in the system of equations for prices of production, given an exogenous specification of the scale factor for the rates of profits. Alternatively, the scale factor can be found, given the wage. Points in common (intersections) on the wage curves for the two techniques are switch points.

Depending on parameter values for the markups on the rates of profits, the example can have no, one, or two switch points. In the last case, the model is one of the reswitching of techniques.

A bifurcation diagram partitions the parameter space into regions where the model solutions, throughout a region, are topologically equivalent, in some sense. Theoretically, a bifurcation diagram for the example should be drawn in an eleven-dimensional space. I, however, take the technology as given and only vary the markups. Figure 1, is the resulting bifurcation diagram.

The model exhibits a certain invariance, manifested in the bifurcation diagram by the straight lines through the origin. Suppose each markup for the rates of profits were, say, doubled. Then, if the scale factor for the rates of profits were halved, the rates of profits in each industry would be unchanged. The wage and prices of production would also be unchanged.

So only the ratio between the markups matter for the model solution. In some sense, the two parameters for the markups can be reduced to one, the ratio between the rates of profits in the two industries. And this ratio is constant for each straight line in the bifurcation diagram. The reciprocal of the slopes of the lines labeled 2 and 4 in Figure 1 are approximately 0.392 and 0.938, respectively. These values are marked along the abscissa in the figure at the top of this post.

In the bifurcation diagram in Figure 1, I have numbered the regions and the loci constituting the boundaries between them. In a bifurcation diagram, one would like to know what a typical solution looks like in each region and how bifurcations occur. The point in this example is to understand changes in the relationships between the wage curves for the two techniques. And the wage curves for the techniques for the numbered regions and lines in Figure 1 look like (are topologically equivalent to) the corresponding numbered graphs in Figure 2 in this post

The model of oligopoly being analyzed here is open, insofar as the determinants of the functional distribution of income, of stable relative rates of profits among industries, and of the long run rate of growth have not been specified. Only comparisons of long run positions are referred to in talking about variations, in the solution to a model of prices of production, with variations in model parameters. That is, no claims are being made about transitions to long period equilibria. Nevertheless, the implications of the results in this paper for short period models, whether ones of classical gravitational processes, cross dual dynamics, intertemporal equilibria, or temporary equilibria, are well worth thinking about.

Mainstream economists frequently produce more complicated models, with conjectural variations, or game theory, or whatever, of firms operating in non-competitive markets. And they seem to think that models of competitive markets are more intuitive, with simple supply and demand properties and certain desirable properties. I think the Cambridge Capital Controversy raised fatal objections to this view long ago. Reswitching and capital reversing show that equilibrium prices are not scarcity indices, and the logic of comparisons of equilibrium positions, in competitive conditions does not conform to the principle of substitution. In the model of prices of production discussed here, there is a certain continuity between imperfections in competition and the case of free competition. The kind of dichotomy that I understand to exist in mainstream microeconomics just doesn't exist here.

Tuesday, June 20, 2017

Continued Bifurcation Analysis of a Reswitching Example

Figure 1: Bifurcation Diagram

This post is a continuation of the analysis in this reswitching example. That post presents an example of reswitching in a model of the production of commodities by means of commodities. The example is one of an economy in which two commodities, iron and corn, are produced. Managers of firms know of two processes for producing iron and one process for producing corn. The definition of technology results in a choice between two techniques of production.

The two-commodity model analyzed here is specified by nine parameters. Theoretically, a bifurcation diagram should be drawn in nine dimensions. But, being limited by the dimensions of the screen, I select two parameters. I take the inputs per unit output in the two processes for producing iron as given constants. I also take as given the amount of (seed) corn needed to produce a unit output of corn, in the one process known for producing corn. So the dimensions of my bifurcation diagram are the amount of labor required to produce a bushel corn and the amount of iron input required to produce a bushel corn. Both of these parameters must be non-negative.

I am interested in wage curves and, in particular, how many intersections they have. Figure 1, above, partitions the parameter space based on this rationale. I had to think some time about what this diagram implies for wage curves. In generating the points to interpolate, my Matlab/Octave code generated many graphs analogous to those in the linked post. I also generated Figure 2, which illustrates configurations of wage curves and switch points, for the number regions and loci in Figure 1. So I had some visualization help, from my code, in thinking about these implications. Anyways, I hope you can see that, from perturbations of one example, one can generate an infinite number of reswitching examples.

Figure 2: Some Wage Curves

One can think of prices of production as (not necessarily stable) fixed points of short period dynamic processes. Economists have developed a number of dynamic processes with such fixed points. But I leave my analysis open to a choice of whatever dynamic process you like. In some sense, I am applying bifurcation analysis to the solution(s) of a system of algebraic equations. The closest analogue I know of in the literature is Rosser (1983), which is, more or less, a chapter in his well-known book.

Update (22 Jun 2017): Added Figure 2, associated changes to Figure 1, and text.

References
  • J. Barkley Rosser (1983). Reswitching as a Cusp Catastrophe. Journal of Economic Theory V. 31: pp. 182-193.

Thursday, June 15, 2017

Perfect Competition With An Uncountable Infinity Of Firms

1.0 Introduction

Consider a partial equilibrium model in which:

  • Consumers demand to buy a certain quantity of a commodity, given its price.
  • Firms produce (supply) a certain quantity of that commodity, given its price.

This is a model of perfect competition, since the consumers and producers take the price as given. In this post, I try to present a model of the supply curve in which the managers of firms do not make systematic mistakes.

This post is almost purely exposition. The exposition is concrete, in the sense that it is specialized for the economic model. I expect that many will read this as still plenty abstract. (I wish I had a better understanding of mathematical notation in HTML.) Maybe I will update this post with illustrations of approximations to integrals.

2.0 Firms Indexed on the Unit Interval

Suppose each firm is named (indexed) by a real number on the (closed) unit interval. That is, the set of firms, X, producing the given commodity is:

X = (0, 1) = {x | x is real and 0 < x < 1}

Each firm produces a certain quantity, q, of the given quantity. I let the function, f, specify the quantity of the commodity that each firm produces. Formally, f is a function that maps the unit interval to the set of non-negative real numbers. So q is the quantity produced by the firm x, where:

q = f(x)
2.1 The Number of Firms

How many firms are there? An infinite number of decimal numbers exist between zero and unity. So, obviously, an infinite number of firms exist in this model.

But this is not sufficient to specify the number of firms. Mathematicians have defined an infinite number of different size infinities. The smallest infinity is called countable infinity. The set of natural numbers, {0, 1, 2, ...}; the set of integers, {..., -2, -1, 0, 1, 2, ...}; and the set of rational numbers can all be be put into a one-to-one correspondence. Each of these sets contain a countable infinity of elements.

But the number of firms in the above model is more than that. The firms can be put into a one-to-one correspondence with the set of real numbers. So there exist, in the model, a uncountable infinity of firms.

2.2 To Know

Cantor's diagonalization argument, power sets, cardinal numbers.

3.0 The Quantity Supplied

Consider a set of firms, E, producing the specified commodity, not necessarily all of the firms. Given the amount produced by each firm, one would like to be able to say what is the total quantity supplied by these firms. So I introduce a notation to designate this quantity. Suppose m(E, f) is the quantity supplied by the firms in E, given that each firm in (0, 1) produces the quantity defined by the function f.

So, given the quantity supplied by each firm (as specified by the function f) and a set of firms E, the aggregate quantity supplied by those firms is given by the function m. And, if that set of firms is all firms, as indexed by the interval (0, 1), the function m yields the total quantity supplied on the market.

Below, I consider for which set of firms m is defined, conditions that might be reasonable to impose on m, a condition that is necessary for perfect competition, and two realizations of m, only one of is correct.

You might think that m should obviously be:

m(E, f) = ∫Ef(x) dx

and that the total quantity supplied by all firms is:

Q = m((0,1), f) = ∫(0, 1) f(x) dx

Whether or not this answer is correct depends on what you mean by an integral. Most introductory calculus classes, I gather, teach the Riemann integral. And, with that definition, the answer is wrong. But it takes quite a while to explain why.

3.1 A Sigma Algebra

One would like the function m to be defined for all subsets of (0, 1) and for all functions mapping the unit interval to the set of non-negative real numbers. Consider a "nice" function f, in some hand-waving sense. Let m be defined for a set of subsets of (0, 1) in which the following conditions are met:

  • The empty set is among the subsets of (0, 1) for which m is defined.
  • m is defined for the interval (0, 1).
  • Suppose m is defined for E, where E is a subset of (0, 1). Let Ec be those elements of (0, 1) which are not in E. Then m is defined for Ec.
  • Suppose m is defined for E1 and E2, both being subsets of (0, 1). Then m is defined for the union of E1 and E2.
  • Suppose m is defined for E1 and E2, both being subsets of (0, 1). Then m is defined for the intersection of E1 and E2.

One might extend the last two conditions to a countable infinity of subsets of (0, 1). As I understand it, any set of subsets of (0, 1) that satisfy these conditions is a σ-algebra. A mathematical question arises: can one define the function m for the set of all subsets of (0, 1)? At any rate, one would like to define m for a maximal set of subsets of (0, 1), in some sense. I think this idea has something to do with Borel sets.

3.2 A Measure

I now present some conditions on this function, m, that specifies the quantity supplied to the market by aggregating over sets of firms:

  • No output is produced by the empty set of firms:
  • m(∅, f) = 0.
  • For any set of firms in the sigma algebra, market output is non-negative:
  • m(E, f) ≥ 0.
  • For disjoint sets of firms in the sigma algebra, the market output of the union of firms is the sum of market outputs:
  • If E1E1 = ∅, then m(E1E1, f) = m(E1, f) + m(E2, f)

The last condition can be extended to a countable set of disjoint sets in the sigma algebra. With this extension, the function m is a measure. In other words, given firms indexed by the unit interval and a function specifying the quantity supplied by each firm, a function mapping from (certain) sets of firms to the total quantity supplied to a market by a set of firms is a measure, in this mathematical model.

One can specify a couple other conditions that seem reasonable to impose on this model of market supply. A set of firms indexed by an interval is a particularly simple set. And the aggregate quantity supplied to the market, when each of these firms produce the same amount is specified by the following condition:

Let I = (a, b) be an interval in (0, 1). Suppose for all x in I:

f(x) = c

Then the quantity supplied to the market by the firms in this interval, m(I, f), is (b - a)c.

3.3 Perfect Competition

Consider the following condition:

Let G be a set of firms in the sigma algebra. Define the function fG(x) to be f(x) when x is not an element of G and to be 1 + f(x) when x is in G. Suppose G has either a finite number of elements or a countable infinity number of elements. Then:

m((0,1), f) = m((0,1), fG)

One case of this condition would be when G is a singleton. The above condition implies that when the single firm increases its output by a single unit, the total market supply is unchanged.

Another case would be when G is the set of firms indexed by the rational numbers in the interval (0, 1). If all these firms increased their individual supplies, the total market supply would still be unchanged.

Suppose the demand price for a commodity depends on the total quantity supplied to the market. Then the demand price would be unaffected by both one firm changing its output and up to a countably infinite number of firms changing their output. In other words, the above condition is a formalization of perfect competition in this model.

4.0 The Riemann Integral: An Incorrect Answer

I now try to describe why the usual introductory presentation of an integral cannot be used for this model of perfect competition.

Consider a special case of the model above. Suppose f(x) is zero for all x. And suppose that G is the set of rational numbers in (0, 1). So fG is unity for all rational numbers in (0, 1) and zero otherwise. How could one define ∫(0, 1)fG(x) dx from a definition of the integral?

Define a partition, P, of (0, 1) to be a set {x0, x1, x2, ..., xn}, where:

0 = x0 < x1 < x2 < ... < xn = 1

The rational numbers are dense in the reals. This implies that, for any partition, each subinterval, [xi - 1, xi] contains a rational number. Likewise, each subinterval contains an irrational real number.

Define, for i = 1, 2, ..., n the two following quantities:

ui = supremum over [xi - 1, xi] of fG(x)

li = infimum over [xi - 1, xi] of fG(x)

For the function fG defined above, ui is always one, for all partitions and all subintervals. For this function, li is always zero.

A partition can be pictured as defining the bases of successive rectangles along the X axis. Each ui specifies the height of a rectangle that just includes the function whose integral is being sought. For a smooth function (not our example), a nice picture could be drawn. The sum of the areas of these rectangles is an upper bound on the desired integral. Each partition yields a possibly different upper bound. The Riemann upper sum is the sum of the rectangles, for a given partition:

U(fG, P) = (x1 - x0) u1 + ... + (xn - xn - 1) un

For the example, with a function that takes on unity for rational numbers, the Riemann upper sum is one for all partitions. The Riemann lower sum is the sum of another set of rectangles.

L(fG, P) = (x1 - x0) l1 + ... + (xn - xn - 1) ln

For the example, the Riemann lower sum is zero, whatever partition is taken.

The Riemann integral is defined in terms of the least upper bound and greatest lower bound on the integral, where the upper and lower bounds are given by Riemann upper and lower sums:

Definition: Suppose the infimum, over all partitions of (0, 1), of the set of Riemann upper sums is equal to the supremum, also over all partitions, of the set of Riemann lower sums. Let Q designate this common value. Then Q is the value of the Riemann integral:

Q = ∫(0, 1)fG(x) dx

If the infimum of Riemann upper sums is not equal to (exceeds) the supremum of the Riemann lower sums, then the Riemann integral of fG is not defined.

In the case of the example, the Riemann integral is not defined. One cannot use the Riemann integral to calculate the changed market supply from a countably infinite firms each increasing their output by one unit.

5.0 Lebesque Integration

The Riemann integral is based on partitioning the X axis. The Lebesque integral, on the other hand, is based on partitioning the Y axis, in some sense. Suppose one has some measure of the size of the set in the domain of a function where the function takes on some designated value. Then the contribution to the integral for that designated value can be seen as the product of that value and that size. The integral of a function can then be defined as the sum, over all possible values of the function, of such products.

5.1 Lebesque Outer Measure

Consider an interval, I = (a, b), in the real numbers. The (Lebesque) measure of that set is simply the length of the interval:

m*(I) = b - a

Let E be a set of real numbers. Let {In} be a set of an at most countable infinite number of open intervals such that

E is a subset of ∪ In

In other words, {In} is an open cover of E. The (Lebesque) measure of E is defined to be:

m*(E) = inf [m*(I1) + m*(I2) + ...]

where the infimum is taken over the set of countably infinite sets of intervals that cover E.

The Lebesque measure of any set that is at most countably infinite is zero. So the rational numbers is a set of Lebesque measure zero. So is a set containing a singleton.

A measurable set E can be used to decompose any other set A into those elements of that set that are also in E and those elements that are not. And the measure of A is the sum of the measures of those two set.

If a set is not measurable, there exists some set A where that sum does not hold. Given the axiom of choice non-measurable sets exist. As I understand it, the set of all measurable subsets of the real numbers is a sigma algebra.

5.2 Lebesque Integral for Simple Functions

Let E be a measurable subset of the real numbers. Define the characteristic function, χE(x), for E, to be one, if x is an element of E, and zero, if x is not an element of E.

Suppose the function g takes on a finite number of values {a1, a2, ..., an}. Such a function is called a simple function. Let Ai be the set of real numbers where gi = ai. The function g can be represented as:

g(x) = a1 χA1(x) + ... + an χAn(x)

The integral of such a simple function is:

g(x) dx = a1 m*(A1) + ... + an m*(An)

This definition can be extended to non-simple functions by another limiting process.

5.3 Lebesque Upper and Lower Sums and the Integral

The Lebesque upper sum of a function f is:

UL(E, f) = sup over simple functions gf of ∫Eg(x) dx

One function is greater than or equal to another function if the value of the first function is greater than or equal to the value of the second function for all points in the common domain of the functions. The Lebesque lower sum is:

LL(E, f) = inf over simple functions gf of ∫Eg(x) dx

Suppose the Lebesque upper and lower sums are equal for a function. Denote that common quantity by Q. Then this is the value of the Lebesque integral of the function.

Q = ∫Ef(x) dx

When the Riemann integral exists for a function, the Lebesque integral takes on the same value. The Lebesque integral exists for more functions, however. The statement of the fundamental theorem of calculus is more complicated for the Lebesque integral than it is for the Riemann integral. Royden (1968) introduces the concept of a function of bounded variation in this context.

5.4 The Quantity Supplied to the Market

So the quantity supplied to the market by the firms indexed by the set E, when each firm produces the quantity specified by the function f is:

m(E, f) = ∫Ef(x) dx

where the integral is the Lebesque integral. In the special case, where the firms indexed by the rational numbers in the interval (0, 1) each supply one more unit of the commodity, the total quantity supplied to the market is unchanged:

Q = ∫(0, 1)fG(x) dx = ∫(0, 1)f(x) dx

Here is a model of perfect competition, in which a countable infinity of firms can vary the quantity they produce and, yet, the total market supply is unchanged.

6.0 Conclusion

I am never sure about these sort of expositions. I suspect that most of those who have the patience to read through this have already seen this sort of thing. I learn something, probably, by setting them out.

I leave many questions above. In particular, I have not specified any process in which the above model of perfect competition is a limit of models with n firms. The above model certainly does not result from taking the limit at infinity of the number of firms in the Cournot model of systematically mistaken firms. That limit contains a countably infinite number of firms, each producing an infinitesimal quantity - a different model entirely.

I gather that economists have gone on from this sort of model. I think there are some models in which firms are indexed by the hyperreals. I do not know what theoretical problem inspired such models and have never studied non-standard analysis.

Another set of questions I have ignored arises in the philosophy of mathematics. I do not know how intuitionists would treat the multiplication of entities required to make sense of the above. Do considerations of computability apply, and, if so, how?

Some may be inclined to say that the above model has no empirical applicability to any possible actually existing market. The above mathematics is not specific to the economics model. It is very useful in understanding probability. For example, the probability density function for any continuous random variable is only defined up to a set of Lebesque measure zero. And probability theory is very useful empirically.

Appendix: Supremum and Infimum

I talk about the supremum and the infimum of a set above. These are sort of like the maximum and minimum of the set.

Let S be a subset of the real numbers. The supremum of S, written as sup S, is the least upper bound of S, if an upper bound exists. The infimum of S is written as inf S. It is the greatest lower bound of S, if a lower bound exists.

References
  • Robert Aumann (1964). Markets with a continuum of traders. Econometrica, V. 32, No. 1-2: pp. 39-50.
  • H. L. Royden (1968). Real Analysis, second edition.