Saturday, December 31, 2016
Welcome
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.
Friday, February 27, 2015
Bad Math In Good Math
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 MathBesides 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 ProofAnyways, 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:
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 SuggestionsI 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?
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.
Country | Gini Coefficient (Non Provisional) | Year |
Slovenia | 0.245 | 2011 |
Norway | 0.250 | 2011 |
Iceland | 0.251 | 2011 |
Denmark | 0.253 | 2011 |
Czech Republic | 0.256 | 2011 |
Finland | 0.261 | 2012 |
Slovak Republic | 0.261 | 2011 |
Belgium | 0.264 | 2010 |
Sweden | 0.273 | 2011 |
Luxembourg | 0.276 | 2011 |
Netherlands | 0.278 | 2012 |
Austria | 0.282 | 2011 |
Switzerland | 0.289 | 2011 |
Hungary | 0.290 | 2012 |
Germany | 0.293 | 2011 |
Poland | 0.304 | 2011 |
Korea | 0.307 | 2012 |
France | 0.309 | 2011 |
Ireland | 0.312 | 2009 |
Canada | 0.316 | 2011 |
Italy | 0.321 | 2011 |
Estonia | 0.323 | 2011 |
New Zealand | 0.323 | 2011 |
Australia | 0.324 | 2012 |
Greece | 0.335 | 2011 |
Japan | 0.336 | 2009 |
United Kingdom | 0.341 | 2010 |
Portugal | 0.341 | 2011 |
Spain | 0.344 | 2011 |
Israel | 0.377 | 2011 |
United States | 0.389 | 2012 |
Turkey | 0.412 | 2011 |
Mexico | 0.482 | 2012 |
Chile | 0.503 | 2011 |
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.
Monday, February 02, 2015
A Cynical Take By Greece's Finance Minister On Mainstream Economists
I have found Yanis Varoufakis' 2014 book, Economic Indeterminacy: A personnel encounter with economists' peculiar nemesis a bit too abstract for my tastes. I am not sure that game theory counts as a subset of neoclassical economics, although I can see how some game theory meets Varoufakis' definition. One might see how a lot of game theory illustrates the idea that economists, collectively, exhibit weakness of will. That is, a lot of game theory can be used to develop models with multiple equilibria and of nondeterministic outcomes. One might expect economists to shy away from these conclusions.
I find it hard to accept Varoufakis's argument that in games, one might want to deliberately be irrational. I wondered if that was so, wouldn't an opponent see this? And, thus, would not this irrational behavior therefore be rational at a meta-level? Varoufakis' argument is structured to address this objection.
But my point in this post is to quote from the preface:
"...my project's failure was predetermined, at least in the sense that it was never going to cause a shift in the attitudes and demeanour of a profession which operates like a priesthood, dedicated solely to preservation of its dogmas... as well as to the recapitulation of its authority within the universities, the financial sector and the government. Indeed, at no point did I harbour any significant hope that this priesthood would take kindly to the demons of doubt and indeterminacy which my work was bound to give rise to. But it did not matter, at least not at a personal level. My intimate familiarity with the neoclassical models was sufficient to keep me on the roster of neoclassical economics departments, where a capacity to teach these models, and produce academic papers based on them is all that matters.
Looking back at these long years of tampering with, and delving into, the complex models of the neoclassical tradition, I cannot but question my decision to keep pushing, Sisyphus-like, the theoretical rock up the neoclassical hill. Why did I stick to this task, when I knew it would end up in failure? In retrospect, there were two reasons, neither of which was predicated upon any hope of influencing a profession utterly uninterested in the truth status of its models. First, I deeply enjoyed toying with these models as an end-in-itself, just as a clockmaker enjoys taking apart and then re-assembling some old clock for the hell of it. Secondly, and more importantly, I felt it necessary to make it hard for my colleagues to pretend to themselves that the models they were being forced to with, by a particularly authoritarian profession, were logically coherent. Bringing them, even fleetingly, to the point when they had to confess to their models' internal contradictions was, I felt, a victory of sorts; the equivalent of a lone sniper behind enemy lines making life difficult for an army of cocupation." -- Yannis Varoufakis (2004: p. xxiv.)
Varoufakis has some other books that sound interesting and more popular. I think his book; The Global Minotaur: America, Europe and the Future of the Global Economy; might be especially topical at the moment.
Update: Steve Keen provides a link to one exposition of Varoufakis' argument that, in game theory, agents can and will deliberately choose irrational behavior.
Friday, January 23, 2015
Approximating a Continuous Time Markov Process
Figure 1: Rate of Transitions Between States in a Three-State Markov Chain |
This post, about Markov processes, does not have much to do with economics. I here define how to approximate a continuous time Markov chain with a discrete time Markov chain. This mathematics is useful for one way of implementing computer simulations involving Markov chains. That is, I want to consider how to start with a continuous time model and synthesize a realization with a small, constant time step.
2.0 Continuous Time Markov ChainsConsider a stochastic process that, at any non-negative time t is in one of N states. Assume this process satisfies the Markov process: the future history of the process after time t depends only on the state of the process at time t, independently of how the process arrived at that state. I consider here only processes with stationary probability distributions for state transitions and for times between transitions. A continuous time Markov chain is specified by a state transition matrix. In this section, I define such a matrix as well as specifying two additional assumptions.
Formally, let P_{i, J} denote the conditional probability that the next transition will be into state j, given that the process is in state i at time zero. (As seen below, in the notation adopted here it matters that these conditional probabilities are not a function of time.) Assume that for each state, the next transition when the process is in that state is into a different state:
P_{i, i} = 0; i = 0, 1, ..., N - 1
Further, assume that for each state, the time to the next transition is from an exponential distribution with the rate of occurrence of state transitions dependent only on the initial state:
F_{i, j}(t) = 1 - e^{- λi t}; i, j = 0, 1, ..., N - 1;
where F_{i, j}(t) is the conditional probability that the next transition will be before time t, given that the chain is in state i at time zero and that the next transition will be into state j. In other words, F_{i, j}(t) is the Cumulative Distribution Function (CDF) for the specified random variable. Under the above definitions, the stochastic process is a continuous time, finite state Markov chain.
Let P_{i, j}(t) be the conditional probability that the chain is in state j at time t, given that the chain is in state i at time zero. These conditional probabilities satisfy Kolmogorov's forward equation:
,
where the transition rate matrix Q is defined to be:
The elements in each row of the transition rate matrix sum to zero. Kolmogorov's forward equation can be expressed in scalar form:
The above equation applies to continuous time Markov chains with a countably infinite number of states only under certain special conditions.
Steady state probabilities of this Markov chain satisfy:
p Q = 0,
where p is a row vector in which each element is the steady-state probability that the chain is in the corresponding state.
3.0 Discrete Time ApproximationA discrete time Markov chain is specified by a state transition matrix A, where a_{i, j} is the probability that the chain will transition in a time step from state i to state j, given that the chain is in state i at the start of the time step. Steady state probabilities for a discrete time Markov chain satisfy:
p A = p
The above equation compares and contrasts with how steady state probabilities relate to the transition rate matrix in a continuous time Markov chain.
Let the time step h be small enough that the probability of the continuous time Markov chain undergoing two or more transitions in a single time step is negligible. In other words, the following probability, calculated from a Poisson distribution, is close to unity for all states i:
P(0 or 1 transitions in time h | Chain in state i at time 0) =(1 + λ_{i} h) e^{- λi h}
The probability that the chain remains in a given state for a time step is the probability that no transitions occur during that time step, given the state of the chain at the start of the time step. This probability is also found from a Poisson distribution:
a_{i, i} = e^{- λi h} = e^{- qi, i h}; i = 0, 1, ..., N - 1
The probability that the chain transitions to state j, given the chain is in state i at the start of the time step, is the product of:
- The probability that a transition occurs during that time step, and
- The conditional probability that the next transition will be into state j, given the chain is in state i at the start of the time step.
The following equation specifies this probability:
a_{i, j} = (1 - a_{i, i})P_{i, j} = (1 - a_{i, i}) q_{i, j}/(- q_{i, i}); i ≠ j
These equations allow one to write a computer program to synthesize a realization from a finite state Markov chain, given the parameters of a continuous time, finite state Markov chain. Such a program will be based on a discrete time approximation.
4.0 An ExampleConsider a three-state, continuous time Markov chain. Figure 1 shows the rate of transitions between the various states. The transition rate matrix is:
To discretize time, choose a small time step h such that, for all states i, the following probabilities are approximately unity:
P(0 or 1 transitions in time h | Chain in state 0 at time 0) =[1 + (λ_{0, 1} + λ_{0, 2})h] e^{-(λ0, 1 + λ0, 2)h}
P(0 or 1 transitions in time h | Chain in state 1 at time 0) =[1 + (λ_{1, 0} + λ_{1, 2})h] e^{-(λ1, 0 + λ1, 2)h}
P(0 or 1 transitions in time h | Chain in state 2 at time 0) =[1 + (λ_{2, 0} + λ_{2, 1})h] e^{-(λ2, 0 + λ2, 1)h}
The state transition matrix A for the discrete-time Markov chain is:
I have not tested the above with concrete values for a continuous time Markov chain.
Reference- S. M. Ross (1970). Applied Probability Models with Optimization Applications. San Francisco: Holden-Day
Friday, January 16, 2015
Laughing At Neoclassical Economists, Elsewhere
- Matthew Yglesias lists "Nine Things Only Neoclassical Economists Will Understand". Strangely, his twitter announcement of this article is about a tenth.
- Noah Smith purports to explain each thing in only a couple of sentences. Stranegly, only for the Modiliani-Miller theorem does he note, "Obviously this doesn't work in the real world".
- Tyler Cowen attempts to clarify the Heckscher-Ohlin theorem, but fails to note that "capital" cannot be a factor of production in the Heckscher-Ohlin-Samuelson model. (He does note Leontief's empirical demonstration that the theory fails.)