Saturday, December 31, 2016

Welcome

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

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

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

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

Comments Policy: I'm quite lax on enforcing any comments policy. I prefer those who post as anonymous (that is, without logging in) to sign their posts at least with a pseudonym. This will make conversations easier to conduct.

Saturday, March 21, 2015

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 save 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.

Thursday, March 12, 2015

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.

Monday, March 09, 2015

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

Friday, February 27, 2015

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 xr}
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 Da(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?

Thursday, February 19, 2015

What Is A "Special Interest"?

I do not want to compare and contrast analytically precise definitions that answer the question in the post title. (Socrates, as reported by Plato, always asked for a definition after being given examples.) Instead, I give two lists, where I trust the reader to see family resemblances among the items on each list:

  • Ethnic groups like African-Americans; women; the poor; organized labor; and lesbians, gays, bisexuals, and transgenders.
  • Corporations, especially those operating in specific industries (e.g., big oil); Corporate Executive Officers; and owners of small businesses.

I suggest that the policies and culture of a country would be quite different, when the dominant understanding of the phrase, "special interests" was consistent with one or another list.

I think somewhere or other Noam Chomsky has asserted that the second understanding reflects the true meaning or the term, or at least a meaning consistent with what the Founding Fathers of the United States wrote. This quote does not have the look back to classical liberals:

"...these questions have been asked for a long time in polls, a little differently worded so you get some different numbers, but for a long time about half the population was saying, when asked a bunch of open questions - like, Who do you think the government is run for? would say something like that: the few, the special interests, not the people. Now it's 82%, which is unprecedented. It means that 82% of the population don't even think we have a political system, not a small number.

What do they mean by special interests? Here you've got to start looking a little more closely. Chances are, judging by other polls and other sources of information, that if people are asked, Who are the special interests? they will probably say, welfare mothers, government bureaucrats, elitists professionals, liberals who run the media, unions. These things would be listed. How many would say, Fortune 500, I don't know. Probably not too many. We have a fantastic propaganda system in this country. There's been nothing like it in history. It's the whole public relations industry and the entertainment industry. The media, which everybody talks about, including me, are a small part of it. I talk about mostly that sector of the media that goes to a small part of the population, the educated sector. But if you look at the whole system, it's just vast. And it is dedicated to certain principles. It wants to destroy democracy. That's its main goal. That means destroy every form of organization and association that might lead to democracy. So you have to demonize unions. And you have to isolate people and atomize them and separate them and make them hate and fear one another and create illusions about where power is. A major goal of this whole doctrinal system for fifty years has been to create the mood of what is now called anti-politics." -- Noam Chomsky, Class Warfare: Interviews with David Barsamian Common Courage Press (1966): p. 138.

But there is another literature, a post modern literature, that also looks at how people come to associate examples with words. People generally do not think logically, following the rules of predicate calculus. One trying to understand culture should realize this. One might talk about the The politics of the signifier. How does one or another definition, or set of examples, become hegemonic? (For what it is worth, I think Slavoj Zizek is a very intelligent, very well-read, self-aware clown.)

Monday, February 09, 2015

Income Inequality In OECD Countries

I recently took another look at data, available from the Organization for Economic Co-operation and Development (OECD), on income inequality. The Gini coefficient is available on countries in the database, under measures of Social Protection and Well-being. Under that menu, expand the sub menu for Income distribution and poverty, and select inequality. You can see the Gini coefficient (at disposable income, post taxes and transfers) displayed, by country, for various years. Table 1 shows the most recent numbers, sorted from countries with the most equal distribution to the least equal. For one way of thinking about it, the United States is not number 1, since the US is exceeded by Turkey, Mexico, and Chile.

Table 1: Gini Coefficient
CountryGini Coefficient
(Non Provisional)
Year
Slovenia0.2452011
Norway0.2502011
Iceland0.2512011
Denmark0.2532011
Czech Republic0.2562011
Finland0.2612012
Slovak Republic0.2612011
Belgium0.2642010
Sweden0.2732011
Luxembourg0.2762011
Netherlands0.2782012
Austria0.2822011
Switzerland0.2892011
Hungary0.2902012
Germany0.2932011
Poland0.3042011
Korea0.3072012
France0.3092011
Ireland0.3122009
Canada0.3162011
Italy0.3212011
Estonia0.3232011
New Zealand0.3232011
Australia0.3242012
Greece0.3352011
Japan0.3362009
United Kingdom0.3412010
Portugal0.3412011
Spain0.3442011
Israel0.3772011
United States0.3892012
Turkey0.4122011
Mexico0.4822012
Chile0.5032011

The Gini coefficient is a measure of inequality, with a higher Gini coefficient denoting a more unequal distribution of income. It is defined as follows: sort the population in order of increasing income. Plot the percentage of income received by those poorer than each value of income against the percentage of the population with less than that value of income. This is the Lorenz curve, and it will fall below a line with a slope of 45 degrees going through the origin. The Gini coefficient is the ratio of the area between the 45 degree line and the Lorenz curve to the area under the 45 degree line. A Gini coefficient of zero indicates perfect equality, while a Gini coefficient of unity arises when one person receives all income and everybody else gets nothing. Consequently, the Gini coefficient lies between zero and one.