Recurrence Of Capital-Output Ratio Without Reswitching

Figure 1: Recurrence of Capital-Output Ratio

1.0 Introduction

This example is from Arrigo Opocher and Ian Steedman. It illustrates the analysis of an isolated industry in equilibrium. This analysis is therefore more akin to partial equilibrium than to general equilibrium. Sometimes (mainstream?) economists say that the Cambridge Capital Controversies were only about aggregate neoclassical theory, that is, macroeconomics. Or that the CCC has been subsumed by General Equilibrium Theory. The example illustrates that such economists are, as has long been apparent, spouting poppycock.

2.0 Indirect Average Cost Function

Consider a firm that produces widgets from inputs of widgets, unskilled labor, and skilled labor. Let the indirect average cost function be:

c(p, w₁, w₂) = sp + w₁ + w₂

+ 2(pw₁)^1/2 + 2(pw₂)^1/2 + 2γ(w₁w₂)^1/2

where

0 < s < 1

0 < γ

γ ≠ 1

and

• p is the price of a widget. Widgets used as inputs are assumed to be totally consumed in one production period.
• w₁ is the wage for unskilled labor.
• w₂ is the wage for skilled labor.

The indirect average cost function shows the average cost of producing each widget (net), when each firm in the industry is producing the cost-minimizing quantity. That is, each firm is producing at the point where the marginal cost and average cost of production of a widget is the same. Assume all firms face the same indirect average cost function. If a positive rate of (accounting) profit was being earned by any firm, the rate of profit would show up in the arguments of the indirect average cost function for that firm.

This indirect average cost function is homogeneous of the first degree:

c(a p, a w₁, a w₂) = a c(p, w₁, w₂)

This is a conventional assumption for cost functions.

Suppose the firm faces a given price of widgets and given wages for skilled and unskilled labor. By Shephard's lemma, the quantity of each input the firm wants to hire per unit output, given the price of each input, is the derivative of the indirect average cost function with respect to the price of that input. Hence, the capital-output ratio, k(p, w₁, w₂), is:

k(p, w₁, w₂) = ∂c/∂p = s + (w₁/p)^1/2 + (w₂/p)^1/2

Notice that the capital-output ratio is a pure number, unambiguously defined in this example, and independent of prices.

By the same logic, the amount of unskilled labor the managers of the firm desire to hire per widget produced is:

l₁(p, w₁, w₂) = ∂c/∂w₁ = 1 + (p/w₁)^1/2 + γ(w₂/w₁)^1/2

The amount of skilled labor the managers of the firm desire to hire per widget produced is:

l₂(p, w₁, w₂) = ∂c/∂w₂ = 1 + (p/w₂)^1/2 + γ(w₁/w₂)^1/2

The matrix of second derivatives of the indirect average cost function is:

(I am not sure whether it is more common to define the above matrix as the transpose of what I have above.) Anyway, for a positive price of widgets and positive wages, the signs of the second derivatives are as follows:

The signs along the principal diagonal show that the slopes of the per-unit input demand functions slope down. That is, given prices for all but one input, a lower price of that input is associated with a willingness of the firm to employ more of that input per unit produced. The positivity of the off-diagonal elements of the above matrix show that widgets, considered as inputs; unskilled labor; and skilled labor are all substitutes, not complements, in some sense. These signs for the matrix of second derivatives of the indirect average cost function are also conventional properties for cost functions.

3.0 Full Industry Equilibrium

Suppose the industry in which widgets are produced has no barriers to entry or exit. Thus, in the long run, economic profits will have been competed away. For firms to be earning no economic profits, the price of widgets must be equal to the average cost of manufacturing them:

p = c(p, w₁, w₂)

So far, no numeraire has been specified. Let widgets themselves be numeraire. Then:

1 = c(w₁, w₂)

where the argument in the indirect average cost function for widgets has been dropped as otiose.

Consider various levels of w₁, the wage of unskilled labor. For the industry to continue to be in long run equilibrium, the wage of skilled labor, w₂, must vary as well, thereby leaving the average cost of producing a widget as unity. Figure 2 illustrates the resulting wage-wage frontier. (Figures are drawn for s = 1/10 and γ = 2/3.) The highest wage for unskilled labor (when the wage for skilled labor is zero) is ((2 - s)^1/2 - 1)². Since this model is symmetric in skilled and unskilled labor, the highest wage for unskilled labor is likewise ((2 - s)^1/2 - 1)². As long as the rate of accounting profits is zero and technology is given, the wage of unskilled labor can only be higher if the wage of skilled labor is lower.

Figure 2: Wage-Wage Frontier

The wage-wage frontier can be used to find the wage of skilled labor for a given wage of unskilled labor between zero and the maximum. In other words, the frontier is helpful in calculating the ratio of the wage of skilled labor to the wage of unskilled labor, given the wage of unskilled labor. This ratio of wages is independent of the choice of the numeraire.

4.0 Capital and Labor

With the chosen numeraire, the capital-output ratio is:

k(w₁, w₂) = s + (w₁)^1/2 + (w₂)^1/2

Given the wage of unskilled labor, one can find the wage of skilled labor and, consequently, both the ratio of wages of the two types of labor and the capital-output ratio. Figure 1, at the start of this post, graphs the capital-output ratio as the derived function of the ratio of wages.

The capital-output ratio is the same when either skilled or unskilled labor is earning their maximum wage, with the other type of labor being paid a wage of zero. In these two extreme cases, the capital-output ratio is (2 - s)^1/2 - (1 - s). Likewise for any ratio but one of the wage of skilled labor to the wage of unskilled labor between these extremes of zero and infinity, the capital-labor ratio is non-unique. The exception is the ratio of wages at which the function in Figure 1 peaks.

One can see that recurrence of the capital-output ratio is not reswitching. Figures 3 and 4 show, respectively, unskilled labor and skilled labor per unit output as a function of the ratio of wages. As shown in Figure 3, a higher wage of skilled labor accompanied by a lower wage of unskilled labor is associated with firms wanting to employ more unskilled labor per unit output. Likewise, a a higher wage of skilled labor accompanied by a lower wage of unskilled labor is associated with firms wanting to employ less skilled labor per unit output. As far as unproduced inputs go, this example of the isolated firm in long run equilibrium does not contradict outdated and exploded neoclassical intuitions about substitution and the mistaken notion of equilibrium prices as scarcity indices. But, since the functions in Figures 3 and 4 are monotonic, no reswitching of techniques arises in this example.

Figure 3: Unskilled Labor Employed per Unit Output

Figure 4: Skilled Labor Employed per Unit Output

5.0 Conclusion

This post has presented an example of an isolated firm in a long period equilibrium. The indirect average cost function, which includes the cost of the use of an input which itself is produced by the firm's industry, otherwise has utterly conventional properties. The analysis of the firm in a long run equilibrium demonstrates that it is an incoherent thought experiment to consider the equilibrium response of the firm to the variation of one price at a time. Only the variation of more than one price at a time can yield an equilibrium analysis that could be at all empirically relevant.

A result of this analysis is to reveal a non-monotonic response of the capital-output ratio to variations in the relative prices of the two unproduced inputs used by this firm in production. In fact, every possible capital-output ratio, except for one, recurs in the example. This is a step in an argument leading to the conclusion that economic theory is consistent with competitive firms wanting to employ more input per unit output for higher prices of that input, a finding that seems consistent with empirical results.

Election Paradoxes And Faustian Agents

I have been trying to reread Donald Saari on election paradoxes. I have previously considered a few parallels between the Condorcet paradox and models of agents as composed of multiple selves. It seems to me that one could draw more analogies here. I do not plan to pursue the research agenda outlined here - I'm not sure how plausible its results would be. Anyways, Saari provides a comprehensive analysis of a range of voting procedures. Could a fuller range of such procedures - as opposed to pairwise majority rule - be applied to models of multiple selves?

For example, consider a model of a person as having multiple selves, where each one of those selves has a set of preferences over commodities. And suppose the individual, in making choices, resolves those selves with a procedure analogous to an election procedure (e.g., plurality vote, antiplurality vote, Borda Count). Suppose which procedure is used is context-dependent. Can an outside agent modify the context somehow such that the individual follows a different procedure, with consequent effects on the individual's choice?

Or consider two people each composed of the same number of multiple selves, with the preferences of those selves the same across these two people. But suppose each person resolves those selves with a different voting procedure. Maybe these two different voting procedures yield the same "best" choice for one specific menu of choices, but order the non-best choices differently. So if a new menu was created with the best choice removed, these two people - who have identical preferences, in some sense - would make different choices.

I suppose if you follow research along these lines, it would be theoretical research. I do not know how an experiment could elicit the required information to determine the preferences of the multiple selves and the election procedure. I guess the challenge would be to come up with an account consistent with some behavioral anomaly arising in economics experiments. Even better might be to suggest a new experiment and to implement it.

References

• Donald G. Saari (2001). Chaotic Elections! A Mathematician Looks at Voting, AMS.

Bertrand Russell, Crank

On the Post Topic

Some great thinkers compare their work to the works of Nicolaus Copernicus or of Galileo:

"The old logic put thought in fetters, while the new logic gives it wings. It has, in my opinion, introduced the same kind of advance into philosophy as Galileo introduced into physics, making it possible at last to see what kinds of problems may be capable of solution, and what kinds must be abandoned as beyond human powers. And where a solution appears possible, the new logic provides a method which enables us to obtain results that do not merely embody personal idiosyncrasies, but must command the assent of all who are competent to form an opinion." -- Bertrand Russell, Our Knowledge of the External World as a Field For Scientific Method in Philosophy (1914).

"...an imagination better stocked with logical tools would have found a key to unlock the mystery. It is in this way that the study of logic becomes the central study in philosophy: it gives the method of research in philosophy, just as mathematics gives the method in physics. And as physics, which, from Plato to the Renaissance, was as unprogressive, dim, and superstitious as philosophy, became a science through Galileo's fresh observation of facts and subsequent mathematical manipulation, so philosophy, in our own day, is becoming scientific through the simultaneous acquisition of new facts and logical methods.

In spite, however, of the new possibility of progress in philosophy, the first effect, as in the case of physics, is to diminish very greatly the extent of what is thought to be known. Before Galileo, people believed themselves possessed of immense knowledge on all the most interesting questions in physics. He established certain facts as to the way in which bodies fall, not very interesting on their own account, but of quite immeasurable interest as examples of real knowledge and of a new method whose future fruitfulness he himself divined. But his few facts sufficed to destroy the whole vast system of supposed knowledge handed down from Aristotle, as even the palest morning sun suffices to extinguish the stars. So in philosophy: though some have believed one system, and others another, almost all have been of opinion that a great deal was known; but all this supposed knowledge in the traditional systems must be swept away, and a new beginning must be made, which we shall esteem fortunate indeed if it can attain results comparable to Galileo's law of falling bodies." -- Bertrand Russell, ibid.

The "new logic" Russell refers to is set out in, for example, Russell and Whitehead's Principia Mathematica. So Russell is comparing himself to Galileo.

An Approach to a Book Review

I'm glad I read this book, although I think it is basically mistaken. Not surprisingly, given their interactions at Cambridge before World War II, Russell's exposition reminds me of Ludwig Wittgenstein's Tractatus Logico-Philosophicus. Although clearly written, Russell's book has a quite different literary style than Wittgenstein's gnostic utterances and hierarchical structure. Both argue that everyday observations about, say, tables and chairs, should be decomposed into logical conjunctions, negations, and disjunctions of atomic facts, which cannot be further broken down. Russell and Wittgenstein differ on the nature of these atomic facts. For Wittgenstein, the referents for entities in atomic facts are quite mysterious; the specification of what these entities are is not a matter of logic, but of its application. Russell is quite clear that these entities include unintegrated sensations, something like "red patch here now."

Russell outlines how one might combine statements about such entities to construct entities that we see, hear, taste, smell, or feel. He goes on to analyze claims about other minds. The analysis of time leads to comments on Zeno's paradoxes and the mathematical theory of continuity. He also explains the idea of infinity, explaining the then recent theory of Cantor. He tries to present a popular overview of these topics. He acknowledges that some of his exposition is more mathematics than philosophy. But, as you can see above, he thinks previous philosophers and many of his contemporaries stumbled into error because they did not possess these logical and mathematical tools. For later developments along the lines, I gather one can look at such works of logical positivism as Rudolf Carnap's The Logical Structure of the World. I have never read Carnap, but I have read A. J. Ayer's Language, Truth, and Logic.

I recently stumbled somewhere across an argument that Noam Chomsky's approach to linguistics supercedes Russell's application of logic to philosophy. Russell and Chomsky agree that sentences of very different structures can have a close surface appearance, and that the same structure can be exhibited in sentences of different surface appearances. In deciding whether or not propositions are true, or even make sense, one should supposedly concentrate on the meaning captured by this deeper structure. But in trying to analyze the meaning of such propositions as, "The king of France is bald", Russell takes an a priori approach. The adequacy of grammar, however, to characterize sentences in a language is an empirical question. And semantics should be based on the parse trees derived from grammatical analysis of the surface appearances of language, not a logical analysis of the surface appearance. This approach, as I understand it, is analogous to how compilers operate. They apply a semantic analysis to a computer program only after first completing a parsing phase. And Chomsky's approach, I gather, has been influential in Artificial Intelligence.

One can argue that just as Wittgenstein, in Philosophical Investigations, showed his earlier approach in the Tractatus was mistaken, so he also showed Chomsky's approach in linguistics to be mistaken. A fortiori, AI is not possible either. Exposition of the parallelism between Russell and Chomsky's analysis of language makes these claims a bit more clear to me. (I guess Sraffa was not too impressed by Chomsky, either.) I suppose one might look at Norman Malcolm's Wittgenstein: Nothing is Hidden, for a fuller argument against Chomsky along these lines. (I did not get much out of Malcolm when I read him years ago.)

Data By Country On Gross And Net Investment?

My article demonstrating the empirical soundness of a simple labor theory of value needs updating. In particular, I should calculate the rate of profits on total capital. So I need data on both constant and circulating capital, not just circulating capital.

Or, at least, I need data on depreciation expenses by some consistent set of conventions. In other words, I need data on gross and net investment. Perhaps it would be sufficient for empirical approximations to have this data on the country level for every country or region in the world. I do not expect to find such data broken down for each country by industry.

Does anybody have suggestions or comments on where to find such data?

Paul Romer Confused On Capital Theory

I have noted Paul Romer's confusions before. For example, consider the following passage:

"In the conventional specification, total capital K is implicitly defined as being proportional to the sum of all different types of capital. This definition implies that all capital goods are perfect substitutes. One additional dollar of capital in the form of a truck has the same effect on the marginal productivity of mainframe computers as an additional dollar's worth of computers. Equation (1) expresses output as an additively separable function of all the different types of capital goods so that one additional dollar of trucks has no effect on the marginal productivity of computers." -- Paul Romer (1990).

Does Romer think that the so-called factor price curve for all techniques must be an affine function? That price Wicksell effects are always zero? Or maybe he just is trying to buffalo his reader with an ill-thought out use of mathematical symbols.

On his twitter feed, he expresses a disinterest in knowing what he is talking about:

"Sorry, but the capital controversies were a waste of time. No relevance then or now." -- Paul Romer, 16 May 2015, 1:09 PM.

I suppose one might possibly be able to defend this view:

"Economists usually stick to science. Robert Solow (1956) was engaged in science when he developed his mathematical theory of growth. But they can get drawn into academic politics. Joan Robinson (1956) was engaged in academic politics when she waged her campaign against capital and the aggregate production function." -- Paul M. Romer (2015).

One might say Solow was looking to empiricalism when he developed his non-rigorous, loose theory of growth. And, I suppose one could say that some political views were involved in Joan Robinson's insistence that Keynes' theory applies to all runs, both the short run and the long run. And in her attempt to combat the development of pre-Keynesian theories after Keynes, even if such developments were the product of those who called themselves Keynesians in some other context.

But to make such an argument, one would have to have read at least some of Joan Robinson's work from the era. It is clear that Romer has not:

"When I learned mathematical economics, a different equilibrium prevailed. ...when economic theorists used math to explore abstractions, it was a point of pride to do so with clarity, precision, and rigor. Then too, a faction like Robinson’s that risked losing a battle might resort to mathiness as a last-ditch defense, but doing so carried a risk. Reputations suffered.

If we have already reached the lemons market equilibrium where only mathiness is on offer, future generations of economists will suffer... Where would we be now if Robert Solow’s math had been swamped by Joan Robinson’s mathiness?" -- Paul M. Romer (2015).

When, during the Cambridge Capital Controversy, did Robinson try to buffalo readers with pretend rigorous manipulations of imprecisely defined mathematical symbols. How about never? Is never good for you?

Update (21 May 2015): Reactions to Romer from Peter Dorman, Justin Fox, Joshua Gans, Noah Smith, Lars Syll, and Matias Vernengo

Update (24 July 2015): Marc Lavoie and Mario Seccareccia also comment on Romer's confusion.

References

• Romer, Paul M. (1990) Endogenous Technological Change, Journal of Political Economy V. 98, N. 5 (Oct): S71-S102.
• Romer, Paul M. (2015). Mathiness in the Theory of Economic Growth, American Economic Review, V. 105, N. 5: pp. 89-93.

Free Trade In The Popular Consciousness

I think it important to oppose errors, both when they are formulated by supposedly rigorous economic thinkers and when they are popularly repeated. The relationships between ideas on different levels can be complicated.

I like writing about minimum wages because it is a clear case where:

• Textbook teaching is wrong, both on empirical
and on theoretical grounds.
• Until recently at least, surveys of economists showed that they, by and large, accepted the mistaken teaching.

As I understand it, "free trade" is a policy area where economists have even more agreement, based on mistaken theory. (I suppose I should include a link to survey of economists somewhere in this post. Any suggestions?) The book from which the following quote is taken contains a structured literature survey, and is written for the general reader:

"Even if I were wrong about this, and the most sophisticated of mainstream economists did think that there was something flawed about the connection between free trade policies and comparative advantage on its own terms (and not merely where its assumptions have been debased), a further point should be raised... We are here engaged in a project of critiquing the ideological effect generated by comparative advantage - of the rationalization that it provides for certain kinds of policies and socioeconomic arrangements. The views of the most sophisticated of academic economists are one - although not the only - part of this, particularly when a less sophisticated version of their ideas predominate outside the halls of economics departments. The manner in which the ideas of intellectuals permeate education, the media and public debate are often more important to the practices of actual agents, policy-makers, and so on, than the most sophisticated renderings of those ideas that emerge from the academy. It matters little if the most sophisticated of economists doubt the connection between free trade and comparative advantage if politicians, commentators, policy-makers or indeed, the public at large, buy the connection between the two and therefore support free trade policies. Indeed, my point - and this enquiry as a whole - is lent its sharpness by its relevance to the world of action, to policy, to normative concerns, to what people do - and think is right to do - because of the ideas that are peddled by intellectuals." -- Vishaal Kishore (2014). Ricardo's Gauntlet: Economic Fiction and the Flawed Case for Free Trade (Anthem Press).

Growth Versus Levels

As far as I can see, mainstream economists generally believe:

• Policy changes that raise the level of growth of the economy are more important than one over changes to the level of output¹.
• Free trade is a desirable policy.
• This policy preference falls out of the theory of comparative advantage.
• Abolishing protectionist tariffs will result in a one-time increase of the level of output of the economy, but not the rate of growth.

As I understand it, Post Keynesians generally think theories of growth cannot be neatly be separated from theories of business cycles - that is, theories of short run fluctuations in the level of output. This intertwining complicates policy recommendations. Nevertheless, one can look back to Keynes² to see a concern with economic growth.

It is hard to see how the two beliefs in the list above are consistent. Why should mainstream economists these days care much about whether countries put in place so-called free trade agreements? Maybe the question is one of who should govern. Anyways, economists in the public sphere should strive to be clear on what they advocate, and why they do.

Footnotes

1. Robert Lucas famously said (that is, I do not know where), "Once you start thinking about growth, it's hard to think about anything else". I do know that in Lucas (1987), he estimated that consumers "would surrender 42 per cent across the board [in consumption] to obtain an increase in the growth rate from [3 per cent] to [6 per cent." On the other hand, to eliminate aggregate consumption variability of the magnitude seen in the USA from the end of the Second World War to the 1980s would be worth "something less than a tenth of a percentage point" in average consumption.
2. Keynes wrote this paean to economic growth in the midst of the Great Depression:

   "The modern age opened; I think, with the accumulation of capital which began in the sixteenth century... From that time until to-day the power of accumulation by compound interest, which seems to have been sleeping for many generations, was re-born and renewed its strength. And the power of compound interest over two hundred years is such as to stagger the imagination.

   For I trace the beginnings of British foreign investment to the treasure which Drake stole from Spain in 1580. In that year he returned to England bringing with him the prodigious spoils of the Golden Hind. Queen Elizabeth was a considerable shareholder in the syndicate which had financed the expedition. Out of her share she paid off the whole of England’s foreign debt, balanced her Budget, and found herself with about 40,000 [pounds] in hand... Thus, every 1 [pound] which Drake brought home in 1580 has now become 100,000 [pounds]. Such is the power of compound interest!

   If capital increases, say, 2 per cent per annum, the capital equipment of the world will have increased by a half in twenty years, and seven and a half times in a hundred years. Think of this in terms of material things--houses, transport, and the like.

   At the same time technical improvements in manufacture and transport have been proceeding at a greater rate in the last ten years than ever before in history. In the United States factory output per head was 40 per cent greater in 1925 than in 1919. In Europe we are held back by temporary obstacles, but even so it is safe to say that technical efficiency is increasing by more than 1 per cent per annum compound. There is evidence that the revolutionary technical changes, which have so far chiefly affected industry, may soon be attacking agriculture... In quite a few years-in our own lifetimes I mean-we may be able to perform all the operations of agriculture, mining, and manufacture with a quarter of the human effort to which we have been accustomed.

   ...All this means in the long run that mankind is solving its economic problem. I would predict that the standard of life in progressive countries one hundred years hence will be between four and eight times as high as it is to-day. There would be nothing surprising in this even in the light of our present knowledge."

References

• John Maynard Keynes (1931). "Economic Possibilities for our Grandchildren", in Essays in Persuasion, W. W. Norton.
• Robert E. Lucas, Jr. (1987). Models of Business Cycles, Basil Blackwell.
• Dani Rodrik (4 May 2015). The War of Trade Models

Aulë's Children Or Durin's Folk

I have been reading the Younger Edda. I am not sure if I will finish it. Anyways, when I came to an account of the dwarves, I found I recognized many of the names.

"Then the gods set themselves in their high-seats and held counsel. They remembered how the dwarfs had quickened in the mould of the earth like maggots in flesh. The dwarfs had first been created and had quickened in Ymer’s flesh, and were then maggots; but now, by the decision of the gods, they got the understanding and likeness of men, but still had to dwell in the earth and in rocks. Modsogner was one dwarf and Durin another. So it is said in the Vala's Prophecy:

Then went all the gods,
The all-holy gods,
On their judgment seats,
And thereon took counsel
Who should the race
Of dwarfs create
From the bloody sea
And from Blain’s bones.
In the likeness of men
Made they many
Dwarfs in the earth,
As Durin said.

And these, says the Vala, are the names of the dwarfs:

Nye, Nide,
Nordre, Sudre,
Austre, Vestre,
Althjof, Dvalin,
Na, Nain,
Niping, Dain,
Bifur, Bafur,
Bombor, Nore,
Ore, Onar,
Oin, Mjodvitner,
Vig, Gandalf,
Vindalf, Thorin,
File, Kile,
Fundin, Vale,
Thro, Throin,
Thek, Lit, Vit,
Ny, Nyrad,
Rek, Radsvid.

But the following are also dwarfs and dwell in the rocks, while the above-named dwell in the mould:

Draupner, Dolgthvare,
Hor, Hugstare,
Hledjolf, Gloin,
Dore, Ore,
Duf, Andvare,
Hepte, File,
Har, Siar.

But the following come from Svarin’s How to Aurvang on Joruvold, and from them is sprung Lovar. Their names are:

Skirfer, Virfir,
Skafid, Ae,
Alf, Inge,
Eikinslgalde,
Fal, Froste,
Fid, Ginnar."

The above list includes all but one of the thirteen dwarves who travel with Bilbo Baggins in Tolkien's The Hobbit. These are Thorin, Fili (File), Kile (Kile), Oin, Gloin, Dwalin (Dvalin), Ori (Ore), Dori (Dore), Nori (Nore), Bifur, Bofur (Bafur), and Bombur (Bombor). (Presumably, Tolkien had a different opinion on the translation of the alphabet for the Eddas.) Balin, the remaining one of Tolkien's thirteen, is the name of a knight in Thomas Malory's Le Morte d'Arthur. Dain and Durin are two other dwarf names apparently taken from the Eddas. According to the Younger Edda, Gimle (Gimli?) is the name of a palace in the south of the world. Gandalf, Tolkien's greatest wizard, is also a name on the above list.

An Example With Heterogeneous Labor

Figure 1: "Labor Demand" in the Consumer Goods Industry

1.0 Introduction

In this post, I work through an example created by Arrigo Opocher and Ian Steedman. In this example, circulating capital is represented by machines of one of a continuum of types, and I compare stationary states. Unskilled and skilled workers use the machines to produce corn, along with more machines. The output of machines are needed to sustain production in future periods. In the stationary states, the same rate of profits is earned in all industries with a positive output. In fact, only the special case when the rate of profits is zero is considered here.

The (slice of) the so-called factor price frontier in this example resembles Paul Samuelson's surrogate production function. Aggregate relationships in this example are "non-perverse". In other words, they do not violate the outdated and exploded intuition of neoclassical microeconomics. The aggregate production function shows positive, but diminishing, marginal returns, in the relevant range, to inputs of factors of production. Lower wages for unskilled labor are associated with capitalists desiring to employ more unskilled labor in the economy overall.

But a perverse relationship arises in the market for corn. Corn is the only consumer good in the example. If capitalists are to want to employ more unskilled labor directly in the production of corn, the wage for unskilled labor must be higher, not lower (Figure 1). If more unskilled labor is available for production, and markets clear, more corn is produced. But when capitalists choose the cost-minimizing technology, at prices and wages they take as given, the quantity of unskilled labor used as input, in the corn industry, per bushel corn produced, decreases. This decrease overwhelms the increased output of corn, and the employment of unskilled labor in the corn industry declines.

2.0 The Technology

Consider a simple capitalist economy, composed of (unskilled and skilled) workers and capitalists. After replacing (circulating) capital goods, output consists of a single consumption good, corn. Unskilled workers are paid the wage w, and skilled workers are paid the wage W out of the harvest. Both wages are in units of bushels corn per person year. Capitalists obtain the rate of profits r. The technology consists of an infinite number of Constant-Returns-to-Scale (CRS) techniques, indexed by s. Table 1 presents the coefficients of production for a single technique.

Table 1: Inputs Required Per Unit Outputs

Inputs	Machine Industry	Corn Industry
Unskilled Labor	a(s) l(s) Person-Years	l(s) Person-Years
Skilled Labor	a(s) t(s) Person-Year	t(s) Person-Years
Machines	a(s) Machines	1 Machine
Outputs	1 Machine	1 Bushel Corn

Notice that the first column of inputs in Table 1 is proportional to - that is, a constant multiple of the - second column. This is akin to Karl Marx's assumption of a constant Organic Composition of Capital, an unrealistic assumption that simplifies price theory.

The index s for the technology is chosen from a set of real numbers, with √ 6  ≤ s ≤ 3. The parameters of a technique are defined in terms of the index as follows:

a(s) = 2 - (6/s) + (6/s²)

l(s) = 1/s

t(s) = 1/s²

Each different value of the index s is associated with the use of a different type of machine. And different quantites of unskilled and skilled labor must be used with each different type of machine to produce the output.

I compare stationary states under these assumptions:

• L person-years of unskilled labor are available for employment in the economy, with √ 6  ≤ L ≤ 3.
• T = 1 person-years of skilled labor are available for employment in the economy.
• r = 0% is the rate of profits in the stationary states considered here.
• The markets for skilled and unskilled labor both clear.
• The production of machines and corn are adapted to a stationary state. So the endowments of machines (by type) are found by solving the model, not givens.

3.0 Quantity Flows for a Given Technique

Given the type of machine, suppose the quantity of corn, c(s), produced is:

c(s) = [1 - a(s)]/t(s) = s² [1 - a(s)]

Let the number of machines, m(s), produced be:

m(s) = 1/t(s) = s²

Table 2 shows the output of the machine and corn industries, scaled to produce these gross outputs.

Table 2: Quantity FLows

Inputs	Machine Industry	Corn Industry
Unskilled Labor	s a(s) Person-Years	s [1 - a(s)] Person-Years
Skilled Labor	a(s) Person-Year	[1 - a(s)] Person-Years
Machines	s2 a(s) Machines	s2 [1 - a(s)] Machines
Outputs	m(s) Machines	c(s) Bushels Corn

For these quantity flows, the total employment of unskilled labor is s. The total employment of skilled labor is one person-year. The total inputs of machines, which are used up each year, are replaced by the output of the machine industry.

4.0 Stationary State Prices in the Special Case

Section 3 specifies quantity flows in a stationary state, given the type of machine. The capitalists choose the technique, including the machine, based on price. Let corn be numeraire, and suppose workers are paid at the end of the production period. If the same rate of profits is earned in the production of machines and corn, the following pair of equations must be satisfied for the technique in use:

p a(s)(1 + r) + a(s) l(s) w + a(s) t(s) W = p

p(1 + r) + l(s) w + t(s) W = 1

These equations have two degrees of freedom. One is eliminated by only considering the special case in which the rate of profits is zero. The other can be seen by expressing the solution as a function of, say, the wage for unskilled labor. In this sense, the solution of the system of equations for prices in a stationary state, given the special case assumption and the technique, is:

p = a(s)

W = [1 - a(s) - l(s) w]/t(s)

Or:

p = 2 - (6/s) + (6/s²)

W = - s² + s(6 - w) - 6

The wage of skilled labor, given the technique, is an affine function of the wage of unskilled labor. Figure 2 illustrates this function for three different techniques. This figure is akin to Figure 2b on page 197 of Samuelson (1962), which shows how to construct the so-called factor price frontier for Samuelson's surrogate production function.

Figure 2: Wage-Wage Curves

In a stationary state, capitalists will have adopted the cost-minimizing technique. The cost-minimizing technique, given the wage of unskilled labor, corresponds to the technique on the outer envelope (that is, the frontier) formed from all (uncountably infinite) functions that one might plot in Figure 2. One can find the technique on the frontier by setting the derivative, with respect to the index s, of the wage-wage curve equal to zero:

dW/ds = 0

Thus, the machine type used by the cost-minimizing technique, in this special case, is the following function of the wage of unskilled labor:

s = (6 - w)/2

The frontier has the equation:

W = (1/4)w² - 3 w + 3

The wage, w, of skilled labor ranges from 0 to (6 - 2 √ 6 ). The wage of skilled labor, W, ranges from 0 to 3. If the rate of profits were positive, the wage-wage frontier would lie inside the frontier found here.

5.0 Some Aggregate Markets

The results found so far can be combined.

5.1 The Market for Unskilled Labor

I have postulated that L person-years of unskilled labor and one person-year of skilled labor are available for employment in a stationary state. For quantity flows in a stationary state to fully employ both types of labor, the index for the machine type must be:

s = L

For this machine type to correspond to the cost-minimizing technique, given a rate of profits of zero and market clearing for both labor markets, the wage of unskilled labor must be the following function of unskilled labor:

w = 6 - 2 L

Figure 3 plots the wage for unskilled labor, under these assumptions, with the amount of unskilled labor firms want to hire in a stationary state. In this example, for more unskilled labor to be hired in a stationary state, its real wage must be lower. This property is particular to this example; it does not generalize.

Figure 3: Employment of Unskilled Labor

5.2 The Market for Skilled Labor

The analysis so far has shown how to determine the cost minimizing technique and the wage for unskilled labor as a function of the amount of unskilled labor employed in a stationary state. And the wage for skilled labor is a function of the wage for unskilled labor, as shown by the wage-wage frontier. The wage for skilled labor can accordingly be expressed as a function of the amount of unskilled labor employed in a stationary state.

W = L² - 6

Figure 4 shows the wage of skilled labor plotted against the quantity of skilled labor firms desire to hire in this example. In some sense, this function neither slopes up nor down.

Figure 4: Employment of Skilled Labor

5.1 The Market for Capital

Under the above assumptions, one can find the type and number of machines, m(s), produced in a stationary state. For stationary states in which different quantities of unskilled labor are employed, different types of machines will be produced. Quantities of different types of machines are incommensurable; physical measures of different types of capital cannot be plotted together on the same axis. A numeraire measure of the quantity of capital, k, can be found by taking the product of the price of machines and their physical quantity:

K = p m(s) = a(s) m(s)

Under the assumption that markets for unskilled and skilled labor clear, one can express numeraire units of capital as a function of the person-years of unskilled labor employed in a stationary state.

K = 2(L² - 3 L + 3)

Figure 5 shows the rate of profits plotted against the above quantity of capital. In this special case, the rate of profits of capital is a non-increasing function of the quantity of capital.

Figure 5: Value of Capital

6.0 Employment in the Corn Industry

The previous section shows that no phenomena that violates outdated neoclassical price theory arises in aggregate markets for unskilled labor, skilled labor, or capital, in this particular example. But consider how much unskilled labor firms, under these assumptions, want to employ in the production of corn. Figure 1 shows the graph of the wage, w, for unskilled labor against the unskilled labor, l₂, hired in the production of corn. That function can be found as:

l₂ = (-L² + 6 L - 6)/L

And this function slopes up, contrary to what neoclassical economists would have expected about half a century ago.

7.0 Conclusion

If you work through enough examples in production theory, you ought to conclude that it is hard to find any justification for mainstream theories in microeconomics. Why so many economists continue to teach archaic balderdash, and (mis)train their intuition accordingly, is a question.

References

• Arrigo Opocher and Ian Steedman (2013). Unconventional results with surrogate production functions Global and Local Economic Review, V. 17, No. 1: pp. 45-53.
• Paul A. Samuelson (1962). Parable and realism in capital theory: The surrogate production function, Review of Economic Studies, V. 29, No. 3: pp. 193-206.

A Plague On Both Your Houses

In a Bloomberg News piece, Noah Smith makes some false claims. I think his mistakes - what Eatwell and Milgate call an imperfectionist view - are widely shared among many macroeconomists. My belief that these mistakes are widely shared is not overthrown, I think, by the confusions put forth in these later posts by Stephen Williamson and Noah Smith, respectively.

First, we have the mistaken belief that in a perfect world, capitalist economies would move quickly towards equilibrium. Smith starts his column with an anecdote:

"One time, at a dinner, I asked a famous macroeconomist: 'So, what really causes recessions?'

His reply came immediately: 'Unexplained shocks to investment.'"

I take this to be an expression of the freshwater view, as embodied in models of Real Business Cycles. Cycles are to be understood as equilibrium paths responding to exogeneous stochastic shocks. Risk exists, but uncertainty does not. Recessions and depressions occur when workers voluntarily decide to take long vacations.

Second, we have mistaken understandings of price theory and how equilibrium is established:

"The market adjusts by the price mechanism. If the cost of something goes up, the price goes up to match. If demand falls, the price drops until the market clears."

I take this to be a claim that equilibrium prices are indices of relative scarcity, a belief shown to be without logical foundation about half a century ago. Ever since Robert Lucas put forth his critique in the 1970s, mainstream macroeconomists have claimed to be developing models with rigorous microfoundations. And those foundations are supposed to be provided by General Equilibrium Theory, in which agents optimize under constraints.

But many macroeconomists seem to be just ignorant of price theory, as experts in GET, such as Frank Hahn explained long ago. In the most rigorous neoclassical theory, with many commodities and many agents, the assumptions do not lead to the conclusion that prices behave that way. Nor do the theorists have a good story about how equilibrium is established. The mathematics used in mainstream macroeconomists does not allow one to find clear statements of assumptions. At least, I am unable to understand what assumptions mainstream economists think they are making on tastes, technology, and endowments in multicommodity models to justify their macroeconomic modeling. I would rather that economists turn to non-equilibrium modeling, a position that I think Robert Lucas still finds incoherent.

Third, suppose you hold that observed fluctuations in employment and output in capitalist economies can hardly be an equilibrium response. If you held the mistaken ideas about price theory that Noah Smith does, you would think that the empirical behavior of economies could only be explained by introducing some imperfection, some failure of competition, some information asymmetry, or some stickiness or slow adjustment into your theory. And given your empirical beliefs, you would think the development of theory in such a direction is a triumph of science:

"But despite these scattered denunciations and grumbles, sticky prices are enjoying a hard-fought place in the sun. The moral of the story is that if you just keep pounding away with theory and evidence, even the toughest orthodoxy in a mean, confrontational field like macroeconomics will eventually have to give you some respect."

But it is not the case that markets, including the labor market, would rapidly clear if only imperfections did not exist in a market economy. For economists to have reached this as a consensus position is a failure of their profession, not an achievement. Business cycles neither need to be explained as an equilibrium phenomenon, nor need sticky prices be invoked to explain the failure of markets to clear.

Is the topic of the above post orthogonal to a debate Paul Krugman overviews? I am of two minds on Krugman's post. I cannot be too hostile to a blog post illustrated with a homoclinic bifurcation. Maybe a solid appreciation of nonlinearity in macroeconomics is associated these days with heterodox, but not necessarily non-mainstream economics.

References

• John Eatwell and Murray Milgate (2011). The Fall and Rise of Keynesian Economics, Oxford University Press.
• Richard M. Goodwin (1990). Chaotic Economic Dynamics, Oxford University Press.
• Murray Milgate (1982). Capital and Employment: A Study of Keynes's Economics, Academic Press.

How To And How Not To Attack Marx's Economics

1.0 Introduction

I am currently reading John Roemer's Free to Lose. I thought I would outline some areas where Marx can be criticized on economic theory, as well as some areas where I do not think he is not so vulnerable. (I do not think I had previously absorbed Roemer's theory of the emergence of classes from an analysis of reproducible equilibrium. But then the Roemer work I know the best is Analytical Foundations of Marxian Economic Theory, which may predate this explanation.) Another motivation is irritation with a series of post here.

2.0 Labor Theory of Prices

For purposes of this post, I put aside the question of whether prices tend to be proportional to labor values. I think Marx rejected this theory, including in the first volume of Capital. He says so, for example, in this passage:

"From the foregoing investigation, the reader will see that this statement only means that the formation of capital must be possible even though the price and value of a commodity be the same; for its formation cannot be attributed to any deviation of the one from the other. If prices actually differ from values, we must, first of all, reduce the former to the latter, in other words, treat the difference as accidental in order that the phenomena may be observed in their purity, and our observations not interfered with by disturbing circumstances that have nothing to do with the process in question. We know, moreover, that this reduction is no mere scientific process. The continual oscillations in prices, their rising and falling, compensate each other, and reduce themselves to an average price, which is their hidden regulator. It forms the guiding star of the merchant or the manufacturer in every undertaking that requires time. He knows that when a long period of time is taken, commodities are sold neither over nor under, but at their average price. If therefore he thought about the matter at all, he would formulate the problem of the formation of capital as follows: How can we account for the origin of capital on the supposition that prices are regulated by the average price, i. e., ultimately by the value of the commodities? I say 'ultimately,' because average prices do not directly coincide with the values of commodities, as Adam Smith, Ricardo, and others believe." -- Karl Marx, Capital, V. 1 (last footnote in Chapter V.)

I take "average price" in the above passage to be referring to what has also been called "such classical terms as 'necessary price', 'natural price', or 'price of production'" (Piero Sraffa, PCMC: p. 9). And Marx is saying that prices of production do not correspond to labor values, even though he is abstracting from this distinction in the first volume of Capital. Others have also asserted that a contradiction in Marx cannot be found here:

"Writers ... like E. Bohm-Bawerk have asserted that there is a contradiction between the analyses of Volumes I and III which is certainly not to be found there unless one reads into them an interpretation different from that which Marx repeatedly emphasized." -- William J. Baumol, "The Transformation of Values: What Marx 'Really' Meant (An Interpretation)" (, V. 12, N. 1 (Mar. 1974): pp. 51-62,

3.0 Heterogeneous Labor Activities

Employees perform many distinct activities in laboring under the direction of capital. I do not think this observation is sufficient, in itself, to hinder the development of a theory organized around labor values. Consider jobs provided by supposedly unskilled labor, such as stocking shelves in a supermarket or working behind the counter in a fast food restaurant. These sort of jobs are often treated as homogenous, both by workers and employers. Workers in one or other such job can transition among them easily enough in times of high employment.

What are jobs that require vastly different levels or types of skills? I do not think this is a problem for Marx as long as relative wages can be treated as stable:

"We suppose labor to be uniform in quality or, what amounts to the same thing, we assume any differences in quality to have been previously reduced to equivalent differences in quantity so that each unit of labor receives the same wage." -- Piero Sraffa, (1960: p. 10).

As far as I can tell, this is a common position among the classical economists, with Adam Smith providing an early explanation of wage differentials.

A problem can arise here, however. Suppose some skills are acquired through an investment, such as paying for higher education. Perhaps there is a tendency for skilled workers to make decisions based on anticipated rates of return. Then, just as Wicksell effects express the dependence of the price of capital goods on distribution, so relative wages would vary with distribution. And labor values would be dependent on prices. One could then express labor value as a vector of different quantities of different types of non-competing workers. But would the assumption that the economy hangs together - e.g., all commodities are basic - work in this case? Or one could make the claim that even skilled labor is heavily produced in the household and outside of firms run for profits. And, thus, calculations of rates of return for acquisition of many skills for the worker are empirically unimportant. (I think I take this objection, as well as the first response, from Ian Steedman.)

4.0 Labor Values Dependent on Choice of Technique

I take labor values as being found from the processes used in production, as expressed in a Leontief input-output matrix and labor coefficients. The components of such matrices and vectors are given in physical units. The analysis of the choice of technique shows that the cost-minimizing technique varies with distribution. So, here too, labor values depend on prices, instead of vice-versa.

Here one could object that the choice of technique is a highly artificial problem, of interest primarily for an internal critique of neoclassical economics. In actuality, firms do not have a choice at any time of processes from a pre-existing menu. Rather technology evolves as a non-reversible process in historical time.

5.0 Volume III Invariants Cannot All Hold

In the above, I have been concentrating mostly on objections to the premises of Marx's economic theory. Let me consider a conclusion. According to Marx, accounting in labor values allows one to identify certain invariants that hold for the economy as a whole. For example, the sum of labor values for gross outputs of industry is equal to the sum of gross outputs, evaluated at prices of production. And the sum of surplus value across industry is equal to the sum of profits. According to Marx, the competition under which prices of production are formed redistributes total surplus values into aliquot quantities distributed to each industry.

Under the traditional analyses of prices of production, Marx was just wrong. For an arbitrary numéraire, not all invariants can simultaneously hold.

Four answers have been given to this issue. I do not think highly of traditional Marxists who argue that one or the other invariant should be given preference. Typically, such arguments are presented with a lot of Hegelian terminology. I find intriguing the argument that all invariants can hold if one adopts Sraffa's standard commodity as the numéraire. Duncan Foley and Gerard Duménil have proposed the new interpretation, organized around the concept of the Monetary Expression of Labor Value (MELT). As I understand it, the new interpretation makes Marx's claims too much a matter of an accounting tautology for my taste. Finally, there is the Temporal Single System Interpretation (TSSI), which I associate mainly with Alan Freeman and Andrew Kliman, although, I guess, they work with many more scholars. Of course, more invariants can be made to hold if you interpret the theory to have many more degrees of freedom.

6.0 Exploitation of Corn

A theorem in the analysis of prices of production states that the rate of profit is positive if and only if labor is exploited. Exploitation here has a technical definition; it is not an ethical concept. From John Roemer, I learn that one can argue that Marx had both ideas in mind.

Anyways, from the same analysis, one can show that same theorem holds for any commodity (that is basic or in the workers' consumption basket?). So why focus on labor? Answers have been given that deal with matters not in the math at this level of abstraction. Workers, unlike owners of commodities sold as means of production, must be brought under the direction of the capitalists when they hire them. Furthermore, the agreements laborers strike are, at best, incomplete contracts. Not all activities that the workers will be expected to perform in given situations can be prespecified. Furthermore, often some will be unpleasant, and a tug-of-war can arise between the worker and the capitalist's representative in the workplace.

Whatever you think of these rationales for focusing on the exploitation of labor, the issue of working conditions seems like a perennial concern.

7.0 Falling Rate of Profit

I do not have much to say about the theory of the falling rate of profit. I think Marx was mistaken here, but recall this is a volume 3 theory, never published in Marx's lifetime. I am aware of Marx's account of countervailing tendencies. (How is this a theory, if no explanation is given why one tendency should predominate?) And, as usual, theorists in the TSSI tradition disagree.

9.0 Outside the Theory of Value and Distribution

Such a brief overview, compared to the thousands of pages Marx wrote, and the many ways scholars and followers have read (parts of?) this work, obviously cannot cover all issues. I have said nothing about historical materialism, for instance. If this theory is read as mandating economic determinism, with no possibility of the superstructure shaping the evolution of the economic base, I, like many others, think the theory is wrong.

Nor have I said anything much about many of Marx's analyses that can be developed independently of the theory of value and distribution. For example, I like to set out Volume 2 models of simple and expanded reproduction in terms of prices of production. Whether or not Richard Goodwin's theory of the business cycle is Marxist or is descriptive of some capitalist economies at some time seems to be independent of Marx's theory of value. And Marx had many other analyses of concrete situations that might or not be worthwhile. For example, in Volume 1, he presented the introduction in Great Britain of laws regulating maximum hours of work as addressing what we would now call a prisoner's dilemma. Each mill owner would like to work their employees until their health breaks, fire them, and then hire refreshed workers. But if all mill owners are doing this for wokers from a young age, no large population of such refreshed workers will exist in the locality. So the owners need such laws after a certain level of development.

I suppose I should say something about the theory of monopoly. I do not see why prices of production cannot be developed with different markups in different industries. I may not be familiar enough with the literature, but it is my impression that many accounts of markup pricing do not take into account constraints arising from the inter-industry flows emphasized in Sraffian theory and empirical work in Leontief input-output analysis. Furthermore, markups cannot be so high in a viable economy that demands total more than the net output of a viable economy. (A theory of cost-push inflation can arise here.) This is not to say that I do not think those exploring administered, full-cost, or markup pricing are not looking at something empirically important.

And Marx had many detailed empirical observations, including claims about how feudalism evolved into capitalism. I cannot address such matters of history. Finally, I have said nothing above about the sociology of economics. I think the above is quite enough for one post.

On Mainstream Economists' Ignorance Of Real Analysis

"Logic sometimes makes monsters. Since half a century we have seen a crowd of bizarre functions which seem to try to resemble as little as possible the honest functions which serve some purpose. No longer continuity, or perhaps continuity, but no derivatives, etc. Nay more, from the logical point of view, it is these strange functions which are the most general, those one meets without seeking no longer appear except as particular cases. There remains for them only a small corner.

Heretofore when a new function was invented, it was for some practical end; to-day they are invented expressly to put at fault the reasonings of our fathers, and one never will get from them anything more than that." -- Henri Poincaré (1908, as quoted in Lakatos 1976, pp. 22-23).

Mainstream economists these days seem unwilling to accept claims about economics that are not backed up by mathematical models. (I think that views on mathematical formalism are pluralistic among non-mainstream economists. Mathematical models are just one of several approaches to acceptable claims about economics, and some non-mainstream economists are quite good at producing mathematical models.) Generally speaking, mainstream economists seem to me to reject norms common among mathematicians.

Anybody taking a standard undergraduate sequence in mathematics at a reasonably good university has an opportunity to be introduced to real analysis. Often, such a class is where the mathematician is introduced to a certain style of definitions and proofs, particularly epsilon-delta proofs. Besides this style, these classes teach a certain content, that is, the theory of limits, the differential calculus, and the integral calculus, from a rigorous standpoint. (I also draw on measure theory below, which, for me, was not taught at the undergraduate level.) In such a class, one should see various examples and purported counter-examples. The examples help the student to understand the range of behavior consistent with certain axioms. The supposed counter-examples help the student understand why theorems contain certain assumptions and why certain concepts useful for stating these assumptions were introduced into mathematics. Given an example inconsistent with the conclusion of a theorem, the student should identify a clause in the assumptions of the theorem that rules out the example.

To make my point, I'll list some examples. For my amusement, I'm not (initially) looking up anything for this post. Just as when someone criticizes somebody else's grammar, the probability approaches unity that they will make a typographic error, so I'll almost certainly be mistaken somewhere below. Does anybody have suggestions for additions to the following list of examples from real analysis?

1. Define a function that is discontinuous at some point.
2. Define a function that is continuous everywhere, but differentiable nowhere.
3. Define a sequence of functions that converges pointwise, but is not uniformly convergent. (Or is it the other way 'round?)
4. Define a function that is Lebesque integrable, but not Riemann integrable.
5. Provide an example of a non-(Lebesque) measurable set.

The style of reasoning introduced in courses on real analysis has been important in economics since, at least, Debreu (1959). And economics provides many examples analogous to the answers to the above problems. Lexicographic preferences can provide an example of a complete order on a commodity space - that is, rational preferences - that cannot be represented by an utility function. Such preferences highlight the need for an assumption on the continuity of preferences, given that the commodity space is a continuum; "rationality" is not sufficient. Menu-dependent preferences suggest the possibility of specifying deeper structures that do and do not allow the construction of binary preference relation providing an order for a commodity space. I suppose the concept of hemi-continuity is proof generated in economics.

Sraffians have also provided many examples not consistent with outdated mainstream teaching. Ian Steedman's work, over the last quarter century, is particularly good on examples illustrating that the Cambridge critique is not exhausted by the possibilities highlighted by reswitching and capital-reversing. As of yet, economists have not specified any general assumption on production processes that rules out these sort of Sraffian examples and yields neoclassical conclusions. Yet many economists - who, I guess, treat their training in mathematics as a hazing ceremony for induction into the brotherhood of economists - proceed as if they have some such theorem.

Obviously, despite my generalization, some economists, both mainstream and non-mainstream understand and accept mathematical analysis. Maybe more mainstream economists understand than my generalization would suggest. The refusal I have seen of economists to accept their own logic may be the manifestation of anti-intellectualism and boundary-patrolling that I think is so common among properly socialized economists. The general public must not come to understand how vulnerable the conclusions of mainstream economists are to slight perturbations in model assumptions. Demonstrations of the failure of the logic in the teaching and public pronouncements of economists must be distracted in blather about credentials or (false?) irrelevancies about empirical results. What economists say in public and what they say in professional seminars need not be consistent. (This is not quite the right link from Dani Rodrik making his point.) I can easily be led to believe that explanation for some behavior I have seen is more a matter of the sociology of economics and less a lack of understanding of mathematics. So, in general, are economists still exhibiting a century-outdated attitude to mathematics?

Answers

1. This is an easy question. For amusement, I'll name a function that exhibits a discontinuity of the second kind, if I correctly remember the terminology. Consider the limit of the following function of the reals as x approaches zero: f(x) = sin(1/x), if x ≠ 0; 0, if x = 0.
2. Various space filling curves provide examples. I think both Hilbert and Sierpinski provide examples.
3. I'm vague on this one, but consider the Fourier series for a square wave, where the value of the square wave at points of discontinuity is the midpoint of the left-hand and right-hand limits. I think mathematicians greeted Fourier's work on functions that were only piecewise continuous with some degree of incredulity.
4. f(x) = 0, for x rational; 1 for x irrational.
5. Consider a decomposition of the real numbers between zero and unity, inclusive, into equivalence classes. For this example, two real numbers in the range are considered equivalent if the difference between them, modulo one, more or less, is a rational number. The axiom of choice allows one to select a real number in each equivalence class. Take the union, with the index set for the union formed by the choice from each equivalence class. The index set contains an infinite number of elements, and the union is the desired closed interval. Furthermore, each equivalence class can be put into a one-to-one correspondence with any other equivalence class. Thus, the measure of each equivalence class must be the same. And these measures must add up to one, since that is the Lebesque measure of the closed interval. But assigning a measure of zero to each equivalence class will not do, and the sum over equivalence claess for any finite measure would be positive infinity. So any equivalence class formed in this way in non-measurable.

References

• Gerard Debreu (1959). Theory of Value: An Axiomatic Analysis of Economic Equilibrium. John Wiley & Sons.
• Imre Lakatos (1976). Proofs and Refutations: The Logic of Mathematical Discovery. Cambridge University Press.
• Walter Rudin (1976). Principles of Mathematical Analysis, Third edition. McGraw-Hill.

Purge of Heterodox Economists Underway at Manitoba?

I stumbled across an article published yesterday in "The students' newspaper of the University of Manitoba". Apparently, the Canadian Association of University Teachers (CAUT) published a report, Report of the Ad Hoc Investigatory Committee into the Department of Economics at the University of Manitoba. They are concerned with the violation, in the economics department, of the academic freedom of professors of economics.

Newton Method, Re-Iterated

Figure 1: Cube Roots Of Unity, Rotated, Newton's Method

I have been re-visiting my program for drawing fractals with Newton's method. Newton's method is an iterative method for finding the roots of non-linear systems of equations. That is, it is used to find zeros of functions. For my purposes, Newton's method can be used to draw fractals, although I was pleased to learn a bit more about methods in numerical analysis. I made various improvements to my program, including the the implementation of:

• More polynomial functions whose zeros are desired.
• Rotations and reflections.
• Two additional iterative methods for root finding.

I was pleased that I had thought to define a Java interface for functions whose zeros were sought. (When one looks at one's own code from a couple years ago, one might as well as be looking at code by somebody else.) Each new function could be added by defining a class implementing this interface. Besides specific functions, I defined a general polynomial, with complex coefficients, that maps complex numbers into complex numbers. I defined rotations and reflections by the transformations to the zeros of this general polynomial. A different strategy would need to be specified if one wanted to create a program for drawing fractals for functions that are not limited to being polynomials.

Halley's method is derived from a second-order Taylor approximation. (Newton's method is derived from a first order approximation.) As nearly, as I can see, Halley's method does not produce as interesting fractals. In implementing the method, I had to review a bit about tensors, since the second derivative of a function mapping the real plane into the real plane is a tensor.

Figure 2: Cube Roots Of Unity, Rotated, Halley's Method

I do not have much of an understanding of the rationale for the Chun-Neta method. I can see that it takes less iterations than Newton or Halley's method, although more calculations per iteration than either of those two methods. (The visual result of less iterations is a lighter color around the roots in the image below, as compared with above.) As I understand it, the black lines in the figure are an artifact of my implementation, probably resulting from dividing by zero.

Figure 3: Cube Roots Of Unity, Rotated, Chun-Neta Method

I conclude with an example from a general polynomial, where I defined roots so that the resulting figures would have no obvious symmetries.

Figure 4: A Fourth Degree Polynomial, Halley's Method

Figure 5: A Fourth Degree Polynomial, Chun-Neta Method

References

• Chun, C. and B. Neta (2011). A new sixth-order scheme for nonlinear equations. Applied Mathematics Letters.
• Scott, Melvin, B. Neta, and C. Chun (2011). Basin attractors for various methods. Applied Mathematics and Computation, V. 218: pp. 2584-2599.
• Yau, Lily and A. Ben-Israel (1998). The Newton and Halley methods for complex roots. American Mathematical Monthly, V. 105: pp. 806-818

Bad Math In Good Math

1.0 Introduction and Overview of the Book

Mark C. Chu-Carroll's blog is Good Math, Bad Math. His book is Good Math: A Geek's Guide to the Beauty of Numbers, Logic, and Computation.

A teenager recently asked me about what math he should learn if he wanted to become a computer programmer or game developer. One cannot recommend a textbook (on discrete mathematics?) to answer this, I think. If you do not mind the errors, this popular presentation will do. I like how it presents the building up of all kinds of numbers from set theory. And the order of this presentation seems right, starting with the natural numbers, but then later providing a set theoretic construction in which the Peano axioms were derived. (I suppose Chu-Carroll could also present a complementary explanation of the need for more kinds of numbers by starting out with the problem of finding roots for polynomial equations in which all coefficients are natural numbers. Eventually, you would get to the claim that an nth degree polynomial with coefficients in the complex numbers has n zeros (some possibly repeating) in the complex numbers.)

The book also has an introduction to the theory of computation, with descriptions of Finite State Machines, lambda calculus, and Turing machines. There is an outline of how the universal Turing machine cannot be improved, in terms of what functions can be computed. It doesn't help to add a second or more tapes. Nor does it help to add a two-dimensional tape. The book concludes with a presentation of a function that cannot be computed by a universal Turing machine. The halting problem, as is canonical, is used for an illustration.

2.0 Bad Math Not In Good Math

Besides being interested in popular presentations of mathematics, I was interested in seeing a book developed from blog posts. Chu-Carroll wisely leaves out a large component of his blog, namely the mocking of silly presentations of bad math. I could not do that with this blog. But there is a contrast here. The bad economics I attempt to counter is presented by supposed leaders of the field and heads of supposed good departments. The bad math Chu-Carroll usually writes about is not being to used to make the world a worse place, to obfuscate and confuse the public, to disguise critical aspects of our society. Rather, it is generally presented by people with less influence than Chu-Carroll or academic mathematicians.

2.1 Not a Proof

Anyways, I want to express some sympathy for why some might find some propositions in mathematics hard to accept. I do not want to argue such nonsense as the idea that Cantor's diagonalization argument fails, by conventional mathematical standards; that different size infinities do not exist; or that 0.999... does not equal 1. Anyways, consider the following purported proof of a theorem.

Theorem:

Proof: Define S by the following:

Then a S is:

Subtract a S from S:

Or:

Thus:

The above was what was to be shown.

Corollary: 0.999... = 1

Proof: First note the following:

Some simple manipulations allow one to apply the theorem:

Or:

That is:

2.2 Comments on the Non-Proof and a Valid Proof

I happen to think of the above supposed proof as a heuristic than I know yields the right answer, sort of. A student, when first presented with the above by an authority, say, in high school, might be inclined to accept it. It seems like symbols are being manipulated in conventional ways.

I do not know that I expect a student to notice how various questions are begged above. What does it mean to take an infinite sum? To multiply an infinite sum by a constant? To take the difference between two infinite sums? To define an infinitely repeating decimal number? But suppose one does ask these questions, questions whose answers are presupposed by the proof. And suppose one is vaguely aware of non-standard analysis. Besides how does inequality in the statement of the theorem arise? One might think the wool is being pulled over one's eyes.

How could one prove that 0.999... = 1? First, one might prove the following by mathematical induction:

Then, after defining what it means to take a limit, one could derive the previously given formula for the infinite geometric series as a limit of the finite sum. (Notice that the restriction in the theorem follows from the proof.) Finally, the claim follows, as a corollary, as shown above.

3.0 Errata and Suggestions

I think that this is the most useful part of this post for Chu-Carroll, especially if this book goes through additional printings or editions.

• p. 7, last line: "(n + 1)(n + 2)/n" should be "(n + 1)(n + 2)/2"
• p. 11, 7 lines from bottom: "our model" should be "our axioms".
• p. 19: Associativity not listed in field axioms.
• p. 20: Since the rational numbers are a field, continuity is not part of the axioms defining a field.
• Sections 2.2 and 3.3: Does the exposition of these constructions already presume the existence of integers and real numbers, respectively?
• p. 21: Shouldn't the definition of a cut be (ignoring that this definition already assumes the existence of the real number r) something like (A, B) where:

A = {x | x rational and x ≤ r}

B = {x | x rational and x > r}

• p. 84, footnote: If one is going to note that exclusive or can be defined in terms of other operations, why not note that one of and or or can be defined in terms of the other and not? Same comment applies to if ... then.
• p. 85, last 2 lines: the line break is confusing.
• p. 95, proof by contradiction of the law of the excluded middle: Is this circular reasoning? Maybe thinking of the proof as being in a meta-language saves this, but maybe this is not the best example.
• p. 97, step 1: Unmatched left parenthesis.
• p. 106: Definition of parent is not provided, but is referenced in the text.
• p. 114, base case: Maybe this should be "partition([], [], [], []).
• p. 130: In definitions of union, intersection, and Cartesian product, logical equivalence is misprinted as some weird character. This misprinting seems to be the case throughout the book (e.g., see pp. 140, 141, and 157).
• p. 133 equation: Right arrow misprinted as ">>".
• Chapter 17: Has anybody proved ZFC consistent? I thought it was the merely the case that nobody has found an inconsistency or can see how one would come about.
• p. 148: Might mention that the order being considered in the well-ordering principle is NOT necessarily the usual, intuitive order.
• p. 148: Drop "larger" in the sentence ending as "...there's a single, unique value that is the smallest positive real number larger!"
• p. 163" "powerset" should be "power set".
• p. 164, line 6: "our choice on the continuum as an axiom" is awkward. How about, "our choice about the continuum hypothesis as an axiom"?
• p. 168, Table 3: g + d = e should be g + d = g.
• p. 171-172: Maybe list mirror symmetry or write, "in addition to mirror symmetry".
• Part VI: Can we have something on the Chomsky hierarchy?
• p. 185; p. 186, Figure 15; p. 193): Labeling state A as a final state is inconsistent with the wording on p. 185, but not the wording on p. 193. On p. 185, write "...that consist of any string containing at least one a, followed by any number of bs."
• p. 190: Would not D_a(ab*) be b*, not ab*?
• p. 223: "second currying example" should be "currying example". No previous example has been presented.
• p. 225, towards bottom of page: I do not understand why α does not appear in formal definition of β.
• p. 229: Suggestion: Refer back to recursion in Section 14.2 or to chapter 18.
• p. 244, 5 lines from bottom: Probably γ should not be used here, since γ was just defined to represent Strings, not a generic type. Same comment goes for α.
• p. 245, last bullet: It seems here δ is being used for the boolean type. On the previous page, β was promised to be used for booleans, as in the first step of the example on the bottom of p. 247.
• p. 249 (Not an error): The reader is supposed to understand what "Intuitionistic logic" means, with no more background than that?
• p. 257: Are the last line of the second paragraph and the last line of the page consistent in syntax?
• Can we have an index?