Saturday, November 26, 2022

An Extensive Rent Example

Figure 1: A Wage Curves and Rent for an Example of Extensive Rent

This post is a rewrite of this. It is the third in a series, with the first here and the second here.

The analysis of the choice of technique in models of extensive rent can be based on the construction of wage curves, even though the outer envelope does not represent the cost-minimizing technique. The orders of fertility and rentability are emphasized here. The order of fertility is defined for specified techniques, in which a single quality of land is used in each technique and that land pays no rent. At a given rate of profits, the qualities of land are ordered by wages, with the most fertile land paying the highest wage. The order of rentability specifies the sequence of different qualities of lands from high rent per acre to low rent per acre. Both orders may vary with the wage or the rate of profits. Table 1 presents coefficients of production for an example.

Table 1: The Coefficients of Production
InputIron IndustryCorn Industry
Type 1 Land049/10000
Type 2 Land0059/1000
Type 3 Land0009/20

How much corn can be produced is constrained by the available quantities of each type of land. Endowments of land and requirements for use must be among the givens to analyze the choice of technique in this example. Suppose one hundred acres of each type of land are available, and net output is somewhere between 321 and 443 bushels of corn. Then all three types of land must be farmed with the parameters specified in Figure 1. One type will be only partially farmed. The iron-producing process must be operated in each of the three economically viable techniques. Table 2 describes which type of lands are fully cultivated and which type of land is left partially fallow in each of the Alpha, Beta, and Gamma techniques.

Table 2:
Type 1Type 2Type 3
AlphaFully farmedFully farmedPartially farmed
BetaPartially farmedFully farmedFully farmed
GammaFully farmedPartially farmedFully farmed

For a given technique, the rent on a type of land only partially farmed is zero since it is not scarce. The wage curves in the left pane in Figure 1 are constructed, for each technique, from the price equations provided by the iron-producing process and the corn-producing process for the land that pays no rent. The choice of technique depends on both income distribution and requirements for use (Quadrio-Curzio 1980). Suppose the rate of profits is taken as given. Then the r–order of efficiency or fertility is the order of the wage curves downwards, until requirements for use are satisfied. Since all three types of land must be somewhat cultivated in the example, the wage frontier is the inner envelope of the wage curves. For the illustrated parameters, Alpha is cost-minimizing, and the order of fertility between the switch points is Type 1, Type 2, and Type 3 lands.

One can calculate the cost of capital goods at the given rate of profits and the cost of labor inputs for corn-producing processes on each of Type 1 and Type 2 lands. Since coefficients of production are specified per bushel corn produced, the revenues from each of these processes, at a unit level, are the same as the process operated on Type 1 land. Rent is the difference between revenues and non-land costs on Type 1 and Type 2 lands. Rent per acre is plotted in the right pane for Figure 1. The order of rentability is the order of lands by rent per acre. The order of rentability is the same as the order of fertility between switch points for the given parameters.

The two fluke switch points in Figure 1 do not lie on the (inner) wage frontier. The maximum rate of profits for the wage curve for Alpha is the maximum rate of profits for this example. One of the switch point for the wage curves for the Beta and Gamma techniques is at this maximum rate of profits. As seen in Figure 2, this is another edge case, a fluke that arises in models of extensive rent. By the way, at other parameter ranges the curves for rent per acre as a function of the rate of profits in the right pane in Figure 1 can intersect. The order of rentability can vary with distribution, and fluke cases in which these curves intersect at a maximum rate of profits or a rate of profits of zero can arise.

Figure 2: The Parameter Space for an Example of Extensive Rent

To the southwest in Figure 2, one switch point between the wage curves for Beta and Gamma has vanished over the wage axis, and the other is at a rate of profits exceeding the maximum rate of profits for Alpha. The order of fertility matches the order of rentability. The northeast is of a reswitching example. For such small perturbations, the order of rentability does not change in this example. Thus, the order of fertility matches the order of rentability only at intermediate rates of profits with reswitching here. The northwest and southeast also illustrate parameters for which the order of fertility varies with the rate of profits. The order of fertility varies from the order of rentability at one or the other extreme, as indicated.

Tuesday, November 22, 2022

Fixed Capital And The Emergence Of Reswitching

Figure 1: A Wage Frontier With A Fluke Switch Point

This post is a rewrite of this, without the attempt to draw a connection to structural economic dynamics. This is the second post in a series, starting with this.

A fluke example with fixed capital illustrates the emergence of the reswitching of techniques. Table 1 presents coefficients of production in a perturbation of an example from Schefold (1980). With the first process, workers, under the direction of mangers of firms, manufacture new machines. The remaining two processes are used to produce corn. The last process requires an input of an old machine, which is jointly produced with corn by the second process. Corn is both a consumption good and a capital good, insofar as it is an input into all three processes.

Table 1: The Coefficients of Production
InputMachine IndustryCorn Industry
One ProcessAnother Process
New Machines010
Old Machines001
New Machines100
Old Machines010

The choice of technique corresponds here to the choice of the economic life of the machine. This lifetime is truncated to one year for the Alpha technique, while the machine is operated for its full physical life of two years under the Beta technique. In a pure fixed capital model, the choice of technique can be analyzed by the construction of the wage frontier. The cost-minimizing technique at a given rate of profits has a wage curve on the outer frontier, as illustrated by Figure 1 for a specified parametrization. Managers of firms are willing to operate the machine for two years for any feasible rate of profits. At the maximum wage or a rate of profits of zero, the Alpha technique is also cost-minimizing. The single switch point is a fluke in two ways. First, it lies on the wage axis. Second, the wage curves are tangent at the switch point.

Figure 3 depicts a part of the parameter space for this example. A thin wedge between two partitions extends to the southeast of the point for the parameters corresponding to Figure 1. At the upper edge of this wedge, the two wage curves for the techniques are tangent at a switch point. The example is of reswitching below this partition and within this wedge. At the lower edge of this wedge, the switch point with the lower rate of profits is on the wage axis.

Figure 2: The Parameter Space for an Example with Fixed Capital

Reswitching, in this example of fixed capital, is connected to the economic life of a machine. The economic life is the full two years here for a low and high rate of profits. Truncation occurs for a range of intermediate rates of profits. The specification of which technique is cost-minimizing can be consistent with vastly different functional distributions of income, with another technique being cost-minimizing for less extreme distributions

The switch point at the higher rate of profits in the reswitching region of the parameter space illustrates capital-reversing. Around this switch point, a lower rate of profits is associated with the adoption of a less capital-intensive cost-minimizing technique. At any rate of profits, inputs into production in a stationary state can be evaluated at prices of production, and these evaluations can be summed for each technique. The ratio of capital per worker, for example, is an index of the capital intensity of a technique. A more capital-intensive technique produces more output per worker, but its adoption is not necessarily encouraged by a lower rate of profits or interest rate (Harris 1973). In other words, a higher wage is associated with the adoption of a technique that requires a greater input of labor per bushel corn produced net throughout the economy. Capital-reversing has been shown to occur in other examples without reswitching on the wage frontier. Harcourt (1972) surveys the controversy in which economists, such as Paul Samuelson and Robert Solow, in Cambridge, Massachusetts, struggled to accept these conclusions drawn by other economists, such as Joan Robinson and Piero Sraffa, at the University of Cambridge.

Consider the region to the southwest in Figure 2. A single switch point exists on the wage frontier. Around this switch point, a lower rate of profits is associated with the adoption of a technique with a greater value of capital per person-year and a greater output per worker. Nevertheless, truncating the operation of the machine for one year is associated with a more capital-intensive technique. The demonstration of the invalidity of Austrian capital theory does not even need the phenomena of reswitching and capital-reversing.

Thursday, November 17, 2022

The Emergence Of The Reverse Substitution Of Labor

Figure 1: A Wage Frontier With Two Fluke Switch Points

This post is a rewrite of this, without the attempt to draw a connection to structural economic dynamics.

This post presents an example with circulating capital alone. Table 1 presents the technology for an economy in which two commodities, iron and corn, are produced. Managers of firms know of one process for producing iron and two for producing corn. Each process is specified by coefficients of production, that is, the required physical inputs per unit output. The Alpha technique consists of the iron-producing process and the first corn-producing process. Similarly, the Beta technique consists of the iron-producing process and the second corn-producing process. At any time, managers of firms face a problem of the choice of technique.

Table 1: The Coefficients of Production
InputIron IndustryCorn Industry
Labora0,1 = 1a0,2α = 16/25a0,2β
Irona1,1 = 9/20a1,2α = 1/625a1,2β
Corna2,1 = 2a2,2α = 12/25a2,2β = 27/400

Two parameters are not given numerical values in this specification of technology. The approach taken here is to examine a local perturbation of parameters in a two-dimensional slice of the higher dimensional parameter space defined by the coefficients of production in particular numeric examples. With wages paid out of the surplus product at the end of the period of production, the wage curves for the two techniques are depicted in Figure 1 for a particular parametrization of the coefficients of production. The Beta technique is cost-minimizing for any feasible distribution of income. If the wage is zero and the workers live on air, the Alpha technique is also cost-minimizing.

A switch point is defined in this model of circulating capital to be an intersection of the wage curves. These switch points, for the particular parameter values illustrated in Figure 1, are fluke cases. Almost any variation in the model parameters destroys their interesting properties. A switch point exists at a rate of profits of -100 percent only along a knife edge in the parameter space (Figure 2). Likewise, a switch point exists on the axis for the rate of profits only along another knife edge. The illustrated example, with two fluke switch points, arises at a single point in the parameter space, where these two partitions intersect.

Figure 2: The Parameter Space for the Reverse Substitution of Labor

Figure 2 depicts a partition of the parameter space around the point with these two fluke switch points. Below the horizontal line, the switch point on the axis for the rate of profits has disappeared below the axis. The Beta technique is cost-minimizing for all feasible non-negative rates of profits. Above this locus, the Alpha technique is cost-minimizing for a low enough wage or a high enough feasible rate of profits.

In the northwest, the switch point at a negative rate of profits occurs at a rate of profits lower than 100 percent. Around the switch point at a positive rate of profits, a lower wage is associated with the adoption of the corn-producing process with a larger coefficient for labor. That is, at a higher wage, employment is lower per unit of gross output in the corn industry.

In the northeast of Figure 2, the switch point for a positive rate of profits exhibits the reverse substitution of labor. Around this switch point, a higher wage is associated with the adoption of a process producing the consumer good in which more labor is employed per unit of gross output. The other switch point exists for a rate of profits between -100 percent and zero. Steedman (2006) presents examples with this phenomenon in models with other structures.

Qualitative changes in the wage frontier exist in the parameter space away from the part graphed in Figure 2. The analysis presented here is of local perturbations of the depicted fluke case.

Tuesday, November 15, 2022

Scholarly Socialists During The Second International

Socialism became a mass movement in many European countries about the time of the heyday of the Second International. Many leaders of these movements and those struggling for leadership produced works of scholarship, albeit often with an activist spirit. I think of, for example:

  • Eduard Bernstein. The Preconditions of Socialism and the Tasks for Social Democracy.
  • Nikolai Bukharin. The Economic Theory of the Leisure Class.
  • Richard B. Day and Daniel F. Guido (eds.). 2018. Responses to Marx's Capital. Brill
  • Rudolph Hilferding. Böhm-Bawerk's Criticism of Marx.
  • Rudolph Hilferding. Finance Capital.
  • Karl Kautsky. The Social Revolution.
  • Karl Kautsky. The Path to Power.
  • Antonio Labriola. Essays on the Materialist Conception of History.
  • Rosa Luxemburg. The Accumulation of Capital.
  • G. V. Plekhanov. The Development of the Monist Theory of History.
  • George Bernard Shaw (ed.). Fabian Essays in Socialism.
  • Georges Sorel. Reflections on Violence.

I do not give copyright dates because I am unsure of the publication dates of some in the original german, italian, or russian. The Day and Guido work is a collections of essays from the time. This is hardly a comprehensive list of the literature of the period. All of these works, published before the October revolution, take Marx as serious and important. The authors were not academics, but I am not sure that there was a solid border between academia and politics at the time. Socialists, in works of scholarship, engaged with those developing the then new-fangled marginalist economic theory.

  • Jukka Gronow. 2016. On the Formation of Marxism. Brill.
  • M. C. Howard and J. E. King. 1989. A History of Marxian Economics, vol. 1. Princeton.
  • Ian Steedman (ed.). 1995. Socialism and Marxism in Economics: 1870 - 1930. Routledge.

Saturday, November 12, 2022

Events Without Probability

1.0 Introduction

In a common model, probability theory assigns a number between zero and one to events. But some events cannot be assigned a probability, even zero. Nobody understands probability, in some sense.

This post is not about economics, although these ideas do have an application in mainstream economics. It is not at all novel. This is one of my favorite proofs in all of math, typically taught sometimes after an introductory mathematical analysis class. My undergraduate class in probability and statistics only alluded to measure theory.

2.0 An Introduction to Probability and a Overview

Suppose one has some sort of repeatable experiment, like rolling a pair of dice, dealing out a five-card poker hand, or spinning a spinner. The set of all possible outcomes is the sample space. Consider a subset of the sample space, such as all rolls in which the pair add up to seven, the hand is a straight flush, or the spinner stops at an angle between zero and 180 degrees. That subset is called an event. If all points in the sample space have the same probability of outcome, the probability is the ratio of the size of the subset for the event to the size of the sample space. (I suppose, more generally, this definition works for non-uniform distributions where the "size" includes weights, so to speak.) Probability is a set function, that is it maps each set in some sort of collection of subsets of the sample space into the real numbers.

This definition works well when the sample space contains a finite number of points. The "size" of a set is then just the number of points in the set. But consider the spinner example, where, with appropriate scaling of angles, the sample space is any real number in the interval [0, 1]. The number of points in a range of scaled angles [a, b] and in the sample space is, in both cases, uncountably infinite. Clearly, one needs some other concept of size here. And the concept of a probability measure provides the needed notion.

The demonstration of the existence of a non-measurable set validates the claim in the introduction. In the proof, the interval [0, 1] is partitioned into an uncountably infinite number of sets, each with only a countably infinite number of points in each set. The axiom of choice is used to "construct" from this partition another partition of the interval into a countably infinite number of sets, each with an uncountably infinite number of points. All of these sets have the same measure, and that measure must add up over the countably infinite number of sets to unity. But that measure can thus be neither zero nor non-szero. So some events exist, given the axiom of choice, without a probability.

3.0 Some Properties of a Measure

I use m(S) to denote the measure of a set. For this post, I consider measures with the following properties:

  • The measure of an interval is merely the length of the interval:
m([a, b]) = b - a
  • The measure of the empty set is zero:
m(∅) = 0
  • The measure of any set S is non-negative. For all S
m(S) ≥ 0
  • The measure of a set is translation invariant. Let S * x denote the set formed by adding x to each element S modulo one. (This definition keeps the translation of a set in the unit interval within the unit interval.) For all S and x:
m(S * x) = m(S)
  • The measure of a set is countably additive. Let S1, S2, S3, ... be a sequence of disjoint sets. That is, for any ij, SiSj = ∅. Then:
m(S1S2S3 ...) = m(S1) + m(S2) + m(S3) + ...

I have above selected the properties of specific kinds of measures. But they are all that is needed for the proof of the existence of unmeasurable sets.

In the spinner example, some non-empty sets have a measure of zero. For example, the probability that the spinner will stop at any given real number in the interval is zero. So is the probability that the spinner will stop at any element in a countably infinite set.

4.0 A Partition of the Unit Interval into Uncountably Infinite Equivalence Classes

Define an equivalence relation as follows. Let two real numbers x1, x2 be equivalent if and only if x2 - x1 is a rational number. Partition the real numbers into equivalence classes by this equivalence relation. The rational numbers is one such equivalence class. The set of real numbers that differ from the square root of two by a rational number is another equivalence class. The set of real numbers differing from the square root of three by a rational number is a third equivalence class. The number π generates a third equivalence class. In fact, there are an uncountable infinite number of equivalence classes, and each such class can be put into a one-to-one mapping to the set of rational numbers.

Consider the collection of sets formed by the intersections of these equivalence classes with the unit interval. This is now a partition of the unit interval. I guess this is easy to understand, as compared to what comes next in this proof.

5.0 An Application of the Axiom of Choice

I now apply the axiom of choice. Let S be a set that contains one and only one point from each of the equivalence classes. Thus, S is a subset of the unit interval containing an uncountably infinite number of points. The difference between any two points in S is an irrational number.

For each rational number r in the unit interval, form the translation S * r. The set of rational numbers in the unit interval is countably infinite. That is, these rational numbers can be ordered in a sequence r1, r2, r3, ...

Corresponding to this sequence is a sequence of sets S*r1, S*r2, S*r3, ... Each one of these sets is a translation of the set S. They all contain an uncountably infinite number of points, and they are all subsets of the unit interval. The intersection of any two of these sets is the empty set. Furthermore every point in the unit interval is in exactly one of these sets.

6.0 Finishing the Proof

The union of these disjoint sets is the unit interval, and the measure of the unit interval is unity. Thus:

m(S*r1S*r2S*r3 ...) = m([0, 1]) = 1 - 0 = 1

By countable additivity:

m(S*r1S*r2S*r3 ...) = m(S*r1) + m(S*r2) + m(S*r3) + ...

By translation invariance:

m(S*r1S*r2S*r3 ...) = m(S) + m(S) + m(S) + ...


1 = m(S) + m(S) + m(S) + ...

Suppose the measure of S were zero. Then we would have proven that zero equals one, an obvious contradiction. But suppose the measure of S were positive. Then the right hand size of the above equality would be infinity, another contradiction. So no measure can be assigned to the set S. In the model of a spinner, the set S represents an event, and no probability can be assigned to this event.

7.0 Conclusion

Has anybody proved the existence of an unmeasurable set with a proof that is not a variation of the above? Can such existence be proven without relying on a proof by contradiction? How would you show that all proofs of such existence must use the axiom of choice? I believe it is also the case that the existence of an unmeasurable set implies the axiom of choice, although I have never seen this proven.

Probabilities cannot be assigned to some events. You will almost certainly never encounter such events in practical applications.

Appendix A. Some Mathematical Background

The above post presumes knowledge that the rationals are countable, that the reals constitute an uncountable infinity, and that an equivalence relation yields equivalence classes that partition a set into nonoverlapping subsets.

A.1 The Rationals are Countable

A countably infinite set can be ordered to be in one-to-one correspondence with the natural numbers, {1, 2, 3, ... }. (Sometimes I begin with zero.) Consider the ordering of positive fractions in Table A-1. The subscripts in parantheses show the order. The fractions are the ratio of the column index to the row index. Obviously, this sequence can be repeated forever. But some rational numbers are repeated. For example, the third fraction in the sequence is 1/2. But r(12) is 2/4. So when enumerating the positive rationals, throw out these repeats. Call the resulting sequence r1, r2, r3, and so on.

Table A-1: Ordering the Rational Numbers
1r(1) = 1/1r(2) = 2/1r(6) = 3/1r(7) = 4/1r(15) = 5/1...
2r(3) = 1/2r(5) = 2/2r(8) = 3/2r(14) = 4/2..
3r(4) = 1/3r(9) = 2/3r(13) = 3/3...
4r(10) = 1/4r(12) = 2/4...
5r(11) = 1/5...

So at least the positive rational numbers are countable. Then consider the sequence 0, r1, -r1, r2, -r2, ... Thus, the rational numbers are countable.

A.2 The Reals are Uncountable

The uncountability of the reals are demonstrated by Cantor's diagonizability argument. It is a proof by contradiction.

I think it easiest to think of the proof with real numbers in binary notation. And it is sufficient to show the real numbers between zero and one are uncountable. Accordingly, suppose somebody claims to have a sequence of all the real numbers between zero and one, as in Table A-2. One constructs a new real number as follows. Let the first digit to the right of the binary point be 0, if the corresponding digit in x1 is 1; 1, if the corresponding digit is 0. Let the second digit to the right of the binary point in this new number be 0, if the corresponding digit in x2 is 1; 1, if the corresponding digit is 0. And so on. Given some arbitary sequence, I have now constructed a new real number that cannot appear in the sequence. For it differs from every number in the sequence by at least one digit. So no such sequence can be constructed for the real numbers.

Table A-2: A Purported Ordering of the Real Numbers Between 0.0 and 1.0

This mathematics demonstrates there are different sizes infinities. Nobody know whether or not there is another size infinity between the rationals and the reals. I am not even sure what this question would mean. But, if you accept that the above makes sense, you probably accept that there are an infinity of sizes of infinity bigger than the reals.

A.3 Equivalence Relations

A relation on a set S is a set of ordered pairs of elements in the set. Informally, the ordered pairs denote the subset of the Cartesian product S x S where the relation is true. Let a ˜ b denote an equivalence relation. That is:

  • The relation is reflexive. For all a in S:
a ˜ a
  • The relation is symmetric. For all a, b in S:
If a ˜ b, then b ˜ a
  • The relation is transitive. For all a, b, c in S:
If a ˜ b and b ˜ c, then a ˜ c

In a sense, an equivalence relation is a generalization of equality. Every equivalence relation generates an equivalence classes, and vice versa. An equivalence class C is a subset of S such that for all a, b in C, a is equivalent to b. Equivalence classes partition a set. Every point in the set is in one equivalence class and only in one equivalence class. Any two equivalence classes are disjoint; their intersection is the empty set.

Thursday, November 03, 2022

External Influences On Academic Economics?

1.0 Introduction

A question has arisen elsewhere. Why, except for an interlude during the post war golden age, has a nineteenth century orthodoxy dominated economics departments and treasury departments around the world? Here I do not investigate the details of this orthodoxy or if it does dominate.

2.0 An Authoritarian Point of View

Some people believe that some are better than others. They want to live in a world where those at the top tell those below what to do, and those below jump.

One might think that it would be hard to find people willing to explicitly articulate these feelings in public. But you can find, if you look, Republican candidates for elected offices saying it was a mistake to allow woment to vote or that interracial marriage should be outlawed.

Some who support plutocracy, maybe unknowingly, would rather claim they are for meritocracy. The extreme distribution of wealth and income in, say, the United States is a difficulty for this view. The rewriting of laws over decades to (p)redistribute income upwards is another inconvenience for this point of view. Advocates for such may be in the grip of a reification in which they naturalize political choices. If a meritocracy was ever momentary established, those at the top could still be expected to try to structure society for their advantage and to attempt to get the best for their spawn.

The reproduction of society is a focus of my favorite schools of economics. Persistent high unemployment and a weak social safety net are useful for sustaining plutocracy. Those at the top want those at the bottom worried about how to feed themselves, not in whether they can participate in governing themselves. Those in middling positions should be economically anxious and worry about falling down. A lack of solidarity between those at your level or with those below is useful for plutocrats. Divisions between workers of various sorts, between races, between men and women, between sexual majorities and minorities are all to be encouraged.

3.0 A Humane Point of View

Human beings do not exist for the economy, but the economy, if it exists, exists for human beings. One assesses how well an economy works by how well it elevates those at the bottom. Are they able to feed, clothe, and shelter themselves? Do they have some share in the necessaries and conveniences of life? Do they participate in improvements brought about by innovations and increases in productivity? Are those at the margins increasingly brought into society?

Before and during the industrial revolution, people needed to work to produce the commodities needed to sustain the population. "For even when we were with you, this we commanded you, that if any would not work, neither should he eat" (2 Thessalonians 10).

This attitude is outdated when productivity is raised so high that, with appropriate distribution, all can have enough. Nor is there are virtue in producing what can be sold on the market. Somewhere, Joan Robinson said something like that the distinction between what can be marketed and what cannot is a technical accident. The expansion of national income provides employment, and, under the current system, employment is a source of income and self-esteem.

Doubtless, in a country with an Universal Basic Income, some would devote to themselves to dissipated living. From the standpoint of political economy, I do not have a problem with this.

But if the economy was structured to serve humanity, many would not feel obligated to spend all their days grubbing for a living. One might have more voluntary neighborhood associations beautifying their area. More young people might organize sports leagues and be playing pickup games. Many would spend more time in community theater or music events. Much more could be done by community groups, charities, and other voluntary civic groups. (In my personal life, I am more a patron or donor for such organizations, mostly not local, than a participant.)

The development of point of views consistent with these ideas and their implementation is and should be a threat to plutocracy.

4.0 Some Speculation

If one looks at the funding of academic economic departments, one can certainly identify promotors of authoritarianism and plutocracy.