tag:blogger.com,1999:blog-267065642015-03-30T10:00:05.592-04:00Thoughts On EconomicsRobert Vienneauhttp://www.blogger.com/profile/14748118392842775431noreply@blogger.comBlogger953125tag:blogger.com,1999:blog-26706564.post-1151835560707333482016-12-31T03:00:00.000-05:002015-01-06T06:09:59.760-05:00WelcomeI 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.<br /><br />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.<br /><br />In general, this blog is abstract, and I think I steer clear of commenting on practical politics of the day.<br /><br />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.<br /><br /><B>Comments Policy:</B> 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.Robert Vienneauhttp://www.blogger.com/profile/14748118392842775431noreply@blogger.com63tag:blogger.com,1999:blog-26706564.post-39855019184365780202015-03-21T15:20:00.000-04:002015-03-23T06:24:07.067-04:00On Mainstream Economists' Ignorance Of Real Analysis<BLOCKQUOTE><P>"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. </P><P>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). </P></BLOCKQUOTE><P>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. </P><P>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. </P><P>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? </P><OL><LI>Define a function that is discontinuous at some point.</LI><LI>Define a function that is continuous everywhere, but differentiable nowhere.</LI><LI>Define a sequence of functions that converges pointwise, but is not uniformly convergent. (Or is it the other way 'round?)</LI><LI>Define a function that is Lebesque integrable, but not Riemann integrable.</LI><LI>Provide an example of a non-(Lebesque) measurable set.</LI></OL><P>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 <A HREF="http://robertvienneau.blogspot.com/2009/03/greek-to-me.html">preferences</A> 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. </P><P>Sraffians have also provided many <A HREF="http://robertvienneau.blogspot.com/search/label/Sraffa%20Effects">examples</A> 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. </P><P>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 <A HREF="http://rodrik.typepad.com/dani_rodriks_weblog/2007/09/deconstructing-.html">link</A> 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? </P><B>Answers</B><OL><LI>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 <I>x</I> approaches zero: <I>f</I>(<I>x</I>) = sin(1/<I>x</I>), if <I>x</I> ≠ 0; 0, if x = 0.</LI><LI>Various space filling curves provide examples. I think both Hilbert and Sierpinski provide examples.</LI><LI>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.</LI><LI><I>f</I>(<I>x</I>) = 0, for <I>x</I> rational; 1 for <I>x</I> irrational.</LI><LI>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.</LI></OL><B>References</B><UL><LI>Gerard Debreu (1959). <I>Theory of Value: An Axiomatic Analysis of Economic Equilibrium</I>. John Wiley & Sons.</LI><LI>Imre Lakatos (1976). <I>Proofs and Refutations: The Logic of Mathematical Discovery</I>. Cambridge University Press.</LI><LI>Walter Rudin (1976). <I>Principles of Mathematical Analysis</I>, Third edition. McGraw-Hill.</LI></UL>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com4tag:blogger.com,1999:blog-26706564.post-15552222830423946342015-03-12T15:19:00.000-04:002015-03-12T15:19:00.470-04:00Purge of Heterodox Economists Underway at Manitoba?<P>I stumbled across an <A HREF="http://www.themanitoban.com/2015/03/caut-national-union-demands-changes-to-economics-department/23176/">article</A> published yesterday in "The students' newspaper of the University of Manitoba". Apparently, the Canadian Association of University Teachers (CAUT) published a <A HREF="http://www.umfa.ca/pdf/miscellaneous/CAUTreport_economics_investigation.pdf">report</A>, <I>Report of the Ad Hoc Investigatory Committee into the Department of Economics at the University of Manitoba</I>. They are concerned with the violation, in the economics department, of the academic freedom of professors of economics. </P>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com0tag:blogger.com,1999:blog-26706564.post-91352440778800815742015-03-09T08:02:00.000-04:002015-03-09T08:02:00.141-04:00Newton Method, Re-Iterated<TABLE CELLSPACING="1" CELLPADDING="1" BORDER="0" ALIGN="center"><TR><TD ALIGN="center"><a href="http://2.bp.blogspot.com/-O5AM0JulPM4/VPbuDetM9fI/AAAAAAAAAhc/H_UmTANyv7A/s1600/CubeRootsRotatedNewton.jpg" imageanchor="1" ><img border="0" src="http://2.bp.blogspot.com/-O5AM0JulPM4/VPbuDetM9fI/AAAAAAAAAhc/H_UmTANyv7A/s320/CubeRootsRotatedNewton.jpg" /></a></TD></TR><TR><TD ALIGN="center"><B>Figure 1: Cube Roots Of Unity, Rotated, Newton's Method</B></TD></TR></TABLE><P>I have been re-visiting my program for drawing fractals with <A HREF="http://robertvienneau.blogspot.com/2010/09/nonergodicity-and-butterfly-effect.html">Newton's method</A>. 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: </P><UL><LI>More polynomial functions whose zeros are desired.</LI><LI>Rotations and reflections.</LI><LI>Two additional iterative methods for root finding.</LI></UL><P>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. </P><P><A HREF="http://mathworld.wolfram.com/HalleysMethod.html">Halley's method</A> 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. </P><TABLE CELLSPACING="1" CELLPADDING="1" BORDER="0" ALIGN="center"><TR><TD ALIGN="center"><a href="http://1.bp.blogspot.com/-EV0Ftgu5tv0/VPbtMKvjDlI/AAAAAAAAAhQ/aTilFfTgMFI/s1600/CubeRootsRotatedHalleyBlowup.jpg" imageanchor="1" ><img border="0" src="http://1.bp.blogspot.com/-EV0Ftgu5tv0/VPbtMKvjDlI/AAAAAAAAAhQ/aTilFfTgMFI/s320/CubeRootsRotatedHalleyBlowup.jpg" /></a></TD></TR><TR><TD ALIGN="center"><B>Figure 2: Cube Roots Of Unity, Rotated, Halley's Method</B></TD></TR></TABLE><P>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. </P><TABLE CELLSPACING="1" CELLPADDING="1" BORDER="0" ALIGN="center"><TR><TD ALIGN="center"><a href="http://1.bp.blogspot.com/-5VbL0k1N9yA/VPbuQrDwAlI/AAAAAAAAAhk/OwHlfNai4Zo/s1600/CubeRootsRotatedChunNetaBlowup.jpg" imageanchor="1" ><img border="0" src="http://1.bp.blogspot.com/-5VbL0k1N9yA/VPbuQrDwAlI/AAAAAAAAAhk/OwHlfNai4Zo/s320/CubeRootsRotatedChunNetaBlowup.jpg" /></a></TD></TR><TR><TD ALIGN="center"><B>Figure 3: Cube Roots Of Unity, Rotated, Chun-Neta Method</B></TD></TR></TABLE><P>I conclude with an example from a general polynomial, where I defined roots so that the resulting figures would have no obvious symmetries. </P><TABLE CELLSPACING="1" CELLPADDING="1" BORDER="0" ALIGN="center"><TR><TD ALIGN="center"><a href="http://4.bp.blogspot.com/-E2nvPWXP5OQ/VPbugHipduI/AAAAAAAAAhs/1LoQ589cLFE/s1600/FourthDegreenHalley.jpg" imageanchor="1" ><img border="0" src="http://4.bp.blogspot.com/-E2nvPWXP5OQ/VPbugHipduI/AAAAAAAAAhs/1LoQ589cLFE/s320/FourthDegreenHalley.jpg" /></a></TD></TR><TR><TD ALIGN="center"><B>Figure 4: A Fourth Degree Polynomial, Halley's Method</B></TD></TR></TABLE><TABLE CELLSPACING="1" CELLPADDING="1" BORDER="0" ALIGN="center"><TR><TD ALIGN="center"><a href="http://2.bp.blogspot.com/-eKBwcDt1P-g/VPbut3Fs5VI/AAAAAAAAAh0/5APkKBFYCY0/s1600/FourthDegreenChunNeta.jpg" imageanchor="1" ><img border="0" src="http://2.bp.blogspot.com/-eKBwcDt1P-g/VPbut3Fs5VI/AAAAAAAAAh0/5APkKBFYCY0/s320/FourthDegreenChunNeta.jpg" /></a></TD></TR><TR><TD ALIGN="center"><B>Figure 5: A Fourth Degree Polynomial, Chun-Neta Method</B></TD></TR></TABLE><B>References</B><UL><LI>Chun, C. and B. Neta (2011). A new sixth-order scheme for nonlinear equations. <I>Applied Mathematics Letters</I>.</LI><LI>Scott, Melvin, B. Neta, and C. Chun (2011). Basin attractors for various methods. <I>Applied Mathematics and Computation</I>, V. 218: pp. 2584-2599.</LI><LI>Yau, Lily and A. Ben-Israel (1998). The Newton and Halley methods for complex roots. <I>American Mathematical Monthly</I>, V. 105: pp. 806-818</LI></UL>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com0tag:blogger.com,1999:blog-26706564.post-56938308067271871592015-02-27T15:46:00.000-05:002015-02-27T15:46:00.175-05:00Bad Math In Good Math<B>1.0 Introduction and Overview of the Book</B><P>Mark C. Chu-Carroll's blog is <A HREF="http://www.goodmath.org/blog/">Good Math, Bad Math</A>. His book is <A HREF="http://www.amazon.com/Good-Math-Computation-Pragmatic-Programmers/dp/1937785335/"><I>Good Math: A Geek's Guide to the Beauty of Numbers, Logic, and Computation</I></A>. <P>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 <I>n</I>th degree polynomial with coefficients in the complex numbers has <I>n</I> zeros (some possibly repeating) in the complex numbers.) </P><P>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. </P><B>2.0 Bad Math Not In Good Math</B><P>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. </P><B>2.1 Not a Proof</B><P>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 <A HREF="http://robertvienneau.blogspot.com/2010/08/infinities-of-infinities.html">exist</A>; or that 0.999... does not equal 1. Anyways, consider the following purported proof of a theorem. </P><P><B>Theorem:</B></P><TABLE CELLSPACING="1" CELLPADDING="1" BORDER="0" ALIGN="center"><TR><TD ALIGN="center"><a href="http://2.bp.blogspot.com/-RXmSpJuvFUY/VNo_EGaYPLI/AAAAAAAAAfk/H_-rBh2THqE/s1600/Figure1.jpg" imageanchor="1" ><img border="0" src="http://2.bp.blogspot.com/-RXmSpJuvFUY/VNo_EGaYPLI/AAAAAAAAAfk/H_-rBh2THqE/s320/Figure1.jpg" /></a></TD></TR></TABLE><P><B>Proof:</B> Define <I>S</I> by the following: </P><TABLE CELLSPACING="1" CELLPADDING="1" BORDER="0" ALIGN="center"><TR><TD ALIGN="center"><a href="http://1.bp.blogspot.com/-tvOCaI39gOY/VNo_HaYnMyI/AAAAAAAAAfs/rf3c8OZDJvA/s1600/Figure2.jpg" imageanchor="1" ><img border="0" src="http://1.bp.blogspot.com/-tvOCaI39gOY/VNo_HaYnMyI/AAAAAAAAAfs/rf3c8OZDJvA/s320/Figure2.jpg" /></a></TD></TR></TABLE><P>Then <I>a</I> <I>S</I> is: </P><TABLE CELLSPACING="1" CELLPADDING="1" BORDER="0" ALIGN="center"><TR><TD ALIGN="center"><a href="http://3.bp.blogspot.com/-DA5bW9Bu5ts/VNo_oYuwsnI/AAAAAAAAAf0/DzT3IheXIME/s1600/Figure3.jpg" imageanchor="1" ><img border="0" src="http://3.bp.blogspot.com/-DA5bW9Bu5ts/VNo_oYuwsnI/AAAAAAAAAf0/DzT3IheXIME/s320/Figure3.jpg" /></a></TD></TR></TABLE><P>Subtract <I>a</I> <I>S</I> from <I>S</I>: </P><TABLE CELLSPACING="1" CELLPADDING="1" BORDER="0" ALIGN="center"><TR><TD ALIGN="center"><a href="http://4.bp.blogspot.com/-io4YLMsOdyA/VNpBFwbtW3I/AAAAAAAAAgA/EgXqkTBRq0c/s1600/Figure4.jpg" imageanchor="1" ><img border="0" src="http://4.bp.blogspot.com/-io4YLMsOdyA/VNpBFwbtW3I/AAAAAAAAAgA/EgXqkTBRq0c/s320/Figure4.jpg" /></a></TD></TR></TABLE><P>Or: </P><TABLE CELLSPACING="1" CELLPADDING="1" BORDER="0" ALIGN="center"><TR><TD ALIGN="center"><a href="http://1.bp.blogspot.com/-6sjQ7ewot6Y/VNpBxXCyT1I/AAAAAAAAAgI/vyAxEjpDEII/s1600/Figure5.jpg" imageanchor="1" ><img border="0" src="http://1.bp.blogspot.com/-6sjQ7ewot6Y/VNpBxXCyT1I/AAAAAAAAAgI/vyAxEjpDEII/s320/Figure5.jpg" /></a></TD></TR></TABLE><P>Thus: </P><TABLE CELLSPACING="1" CELLPADDING="1" BORDER="0" ALIGN="center"><TR><TD ALIGN="center"><a href="http://3.bp.blogspot.com/-R-Lje2MLflE/VNpB355neRI/AAAAAAAAAgQ/5yhtL7564gI/s1600/Figure6.jpg" imageanchor="1" ><img border="0" src="http://3.bp.blogspot.com/-R-Lje2MLflE/VNpB355neRI/AAAAAAAAAgQ/5yhtL7564gI/s320/Figure6.jpg" /></a></TD></TR></TABLE><P>The above was what was to be shown. </P><P><B>Corollary:</B> 0.999... = 1 </P><P><B>Proof:</B> First note the following: </P><TABLE CELLSPACING="1" CELLPADDING="1" BORDER="0" ALIGN="center"><TR><TD ALIGN="center"><a href="http://3.bp.blogspot.com/-vCa2OWxb2AI/VNpCy8svuVI/AAAAAAAAAgc/n5S6VrcC878/s1600/Figure7.jpg" imageanchor="1" ><img border="0" src="http://3.bp.blogspot.com/-vCa2OWxb2AI/VNpCy8svuVI/AAAAAAAAAgc/n5S6VrcC878/s320/Figure7.jpg" /></a></TD></TR></TABLE><P>Some simple manipulations allow one to apply the theorem: </P><TABLE CELLSPACING="1" CELLPADDING="1" BORDER="0" ALIGN="center"><TR><TD ALIGN="center"><a href="http://2.bp.blogspot.com/-9ONKbOrtNWc/VNpDzNH0MjI/AAAAAAAAAgo/aEUt2y47amM/s1600/Figure8.jpg" imageanchor="1" ><img border="0" src="http://2.bp.blogspot.com/-9ONKbOrtNWc/VNpDzNH0MjI/AAAAAAAAAgo/aEUt2y47amM/s320/Figure8.jpg" /></a></TD></TR></TABLE><P>Or: </P><TABLE CELLSPACING="1" CELLPADDING="1" BORDER="0" ALIGN="center"><TR><TD ALIGN="center"><a href="http://3.bp.blogspot.com/-9pgvXYcWjis/VNpD3AFHPEI/AAAAAAAAAgw/Io7J1DNXW8Y/s1600/Figure9.jpg" imageanchor="1" ><img border="0" src="http://3.bp.blogspot.com/-9pgvXYcWjis/VNpD3AFHPEI/AAAAAAAAAgw/Io7J1DNXW8Y/s320/Figure9.jpg" /></a></TD></TR></TABLE><P>That is: </P><TABLE CELLSPACING="1" CELLPADDING="1" BORDER="0" ALIGN="center"><TR><TD ALIGN="center"><a href="http://3.bp.blogspot.com/-q292T4pDzAk/VNpD7eVHJOI/AAAAAAAAAg4/F8SyM7GVgTI/s1600/Figure10.jpg" imageanchor="1" ><img border="0" src="http://3.bp.blogspot.com/-q292T4pDzAk/VNpD7eVHJOI/AAAAAAAAAg4/F8SyM7GVgTI/s320/Figure10.jpg" /></a></TD></TR></TABLE><B>2.2 Comments on the Non-Proof and a Valid Proof</B><P>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. </P><P>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. </P><P>How could one prove that 0.999... = 1? First, one might prove the following by mathematical induction: </P><TABLE CELLSPACING="1" CELLPADDING="1" BORDER="0" ALIGN="center"><TR><TD ALIGN="center"><a href="http://1.bp.blogspot.com/-WnDEzT1eSxc/VNpEC33cZUI/AAAAAAAAAhA/n2Xyzu5uX24/s1600/Figure11.jpg" imageanchor="1" ><img border="0" src="http://1.bp.blogspot.com/-WnDEzT1eSxc/VNpEC33cZUI/AAAAAAAAAhA/n2Xyzu5uX24/s320/Figure11.jpg" /></a></TD></TR></TABLE><P>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. </P><B>3.0 Errata and Suggestions</B><P>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><UL><LI>p. 7, last line: "(<I>n</I> + 1)(<I>n</I> + 2)/<I>n</I>" should be "(<I>n</I> + 1)(<I>n</I> + 2)/2"</LI><LI>p. 11, 7 lines from bottom: "our model" should be "our axioms".</LI><LI>p. 19: Associativity not listed in field axioms.</LI><LI>p. 20: Since the rational numbers are a field, continuity is not part of the axioms defining a field.</LI><LI>Sections 2.2 and 3.3: Does the exposition of these constructions already presume the existence of integers and real numbers, respectively?</LI><LI>p. 21: Shouldn't the definition of a <I>cut</I> be (ignoring that this definition already assumes the existence of the real number <I>r</I>) something like (<I>A</I>, <I>B</I>) where:</LI></UL><BLOCKQUOTE><I>A</I> = {<I>x</I> | <I>x</I> rational and <I>x</I> ≤ <I>r</I>} </BLOCKQUOTE><BLOCKQUOTE><I>B</I> = {<I>x</I> | <I>x</I> rational and <I>x</I> > <I>r</I>} </BLOCKQUOTE><UL><LI>p. 84, footnote: If one is going to note that <I>exclusive or</I> can be defined in terms of other operations, why not note that one of <I>and</I> or <I>or</I> can be defined in terms of the other and <I>not</I>? Same comment applies to <I>if</I> ... <I>then</I>.</LI><LI>p. 85, last 2 lines: the line break is confusing.</LI><LI>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.</LI><LI>p. 97, step 1: Unmatched left parenthesis.</LI><LI>p. 106: Definition of <I>parent</I> is not provided, but is referenced in the text.</LI><LI>p. 114, base case: Maybe this should be "partition([], [], [], []).</LI><LI>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).</LI><LI>p. 133 equation: Right arrow misprinted as ">>".</LI><LI>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.</LI><LI>p. 148: Might mention that the order being considered in the well-ordering principle is NOT necessarily the usual, intuitive order.</LI><LI>p. 148: Drop "larger" in the sentence ending as "...there's a single, unique value that is the smallest positive real number larger!"</LI><LI>p. 163" "powerset" should be "power set".</LI><LI>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"?</LI><LI>p. 168, Table 3: g + d = e should be g + d = g.</LI><LI>p. 171-172: Maybe list mirror symmetry or write, "in addition to mirror symmetry".</LI><LI>Part VI: Can we have something on the Chomsky hierarchy?</LI><LI>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 <I>a</I>, followed by any number of <I>b</I>s."</LI><LI>p. 190: Would not <I>D</I><SUB><I>a</I></SUB>(<I>a</I><I>b</I>*) be <I>b</I>*, not <I>a</I><I>b</I>*?</LI><LI>p. 223: "second currying example" should be "currying example". No previous example has been presented.</LI><LI>p. 225, towards bottom of page: I do not understand why α does not appear in formal definition of β.</LI><LI>p. 229: Suggestion: Refer back to recursion in Section 14.2 or to chapter 18.</LI><LI>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.</LI><LI>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.</LI><LI>p. 249 (Not an error): The reader is supposed to understand what "Intuitionistic logic" means, with no more background than that?</LI><LI>p. 257: Are the last line of the second paragraph and the last line of the page consistent in syntax?</LI><LI>Can we have an index?</LI></UL>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com0tag:blogger.com,1999:blog-26706564.post-2760699433882183912015-02-19T15:14:00.000-05:002015-02-19T15:14:00.614-05:00What Is A "Special Interest"?<P>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: </P><UL><LI>Ethnic groups like African-Americans; women; the poor; organized labor; and lesbians, gays, bisexuals, and transgenders.</LI><LI>Corporations, especially those operating in specific industries (e.g., big oil); Corporate Executive Officers; and owners of small businesses.</LI></UL><P>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. </P><P>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: </P><BLOCKQUOTE><P>"...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.</P><P>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, <I>Class Warfare: Interviews with David Barsamian</I> Common Courage Press (1966): p. 138.</P></BLOCKQUOTE><P>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 <A HREF="http://robertvienneau.blogspot.com/2007/03/unconscious-is-structured-like-language.html">politics of the signifier</A>. 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.) </P>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com6tag:blogger.com,1999:blog-26706564.post-77831685951240257792015-02-09T08:14:00.000-05:002015-02-09T08:14:00.117-05:00Income Inequality In OECD Countries<P>I recently took another look at data, available from the Organization for Economic Co-operation and Development (<A HREF="http://stats.oecd.org/">OECD</A>), 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. </P><TABLE CELLSPACING="1" CELLPADDING="1" BORDER="1" ALIGN="center"><CAPTION><B>Table 1: Gini Coefficient</B></CAPTION><TR><TD ALIGN="left"><B>Country</B></TD><TD><B>Gini Coefficient<BR>(Non Provisional)</B></TD><TD><B>Year</B></TD></TR><TR><TD ALIGN="center">Slovenia</TD><TD ALIGN="center">0.245</TD><TD ALIGN="center">2011</TD></TR><TR><TD ALIGN="center">Norway</TD><TD ALIGN="center">0.250</TD><TD ALIGN="center">2011</TD></TR><TR><TD ALIGN="center">Iceland</TD><TD ALIGN="center">0.251</TD><TD ALIGN="center">2011</TD></TR><TR><TD ALIGN="center">Denmark</TD><TD ALIGN="center">0.253</TD><TD ALIGN="center">2011</TD></TR><TR><TD ALIGN="center">Czech Republic</TD><TD ALIGN="center">0.256</TD><TD ALIGN="center">2011</TD></TR><TR><TD ALIGN="center">Finland</TD><TD ALIGN="center">0.261</TD><TD ALIGN="center">2012</TD></TR><TR><TD ALIGN="center">Slovak Republic</TD><TD ALIGN="center">0.261</TD><TD ALIGN="center">2011</TD></TR><TR><TD ALIGN="center">Belgium</TD><TD ALIGN="center">0.264</TD><TD ALIGN="center">2010</TD></TR><TR><TD ALIGN="center">Sweden</TD><TD ALIGN="center">0.273</TD><TD ALIGN="center">2011</TD></TR><TR><TD ALIGN="center">Luxembourg</TD><TD ALIGN="center">0.276</TD><TD ALIGN="center">2011</TD></TR><TR><TD ALIGN="center">Netherlands</TD><TD ALIGN="center">0.278</TD><TD ALIGN="center">2012</TD></TR><TR><TD ALIGN="center">Austria</TD><TD ALIGN="center">0.282</TD><TD ALIGN="center">2011</TD></TR><TR><TD ALIGN="center">Switzerland</TD><TD ALIGN="center">0.289</TD><TD ALIGN="center">2011</TD></TR><TR><TD ALIGN="center">Hungary</TD><TD ALIGN="center">0.290</TD><TD ALIGN="center">2012</TD></TR><TR><TD ALIGN="center">Germany</TD><TD ALIGN="center">0.293</TD><TD ALIGN="center">2011</TD></TR><TR><TD ALIGN="center">Poland</TD><TD ALIGN="center">0.304</TD><TD ALIGN="center">2011</TD></TR><TR><TD ALIGN="center">Korea</TD><TD ALIGN="center">0.307</TD><TD ALIGN="center">2012</TD></TR><TR><TD ALIGN="center">France</TD><TD ALIGN="center">0.309</TD><TD ALIGN="center">2011</TD></TR><TR><TD ALIGN="center">Ireland</TD><TD ALIGN="center">0.312</TD><TD ALIGN="center">2009</TD></TR><TR><TD ALIGN="center">Canada</TD><TD ALIGN="center">0.316</TD><TD ALIGN="center">2011</TD></TR><TR><TD ALIGN="center">Italy</TD><TD ALIGN="center">0.321</TD><TD ALIGN="center">2011</TD></TR><TR><TD ALIGN="center">Estonia</TD><TD ALIGN="center">0.323</TD><TD ALIGN="center">2011</TD></TR><TR><TD ALIGN="center">New Zealand</TD><TD ALIGN="center">0.323</TD><TD ALIGN="center">2011</TD></TR><TR><TD ALIGN="center">Australia</TD><TD ALIGN="center">0.324</TD><TD ALIGN="center">2012</TD></TR><TR><TD ALIGN="center">Greece</TD><TD ALIGN="center">0.335</TD><TD ALIGN="center">2011</TD></TR><TR><TD ALIGN="center">Japan</TD><TD ALIGN="center">0.336</TD><TD ALIGN="center">2009</TD></TR><TR><TD ALIGN="center">United Kingdom</TD><TD ALIGN="center">0.341</TD><TD ALIGN="center">2010</TD></TR><TR><TD ALIGN="center">Portugal</TD><TD ALIGN="center">0.341</TD><TD ALIGN="center">2011</TD></TR><TR><TD ALIGN="center">Spain</TD><TD ALIGN="center">0.344</TD><TD ALIGN="center">2011</TD></TR><TR><TD ALIGN="center">Israel</TD><TD ALIGN="center">0.377</TD><TD ALIGN="center">2011</TD></TR><TR><TD ALIGN="center">United States</TD><TD ALIGN="center">0.389</TD><TD ALIGN="center">2012</TD></TR><TR><TD ALIGN="center">Turkey</TD><TD ALIGN="center">0.412</TD><TD ALIGN="center">2011</TD></TR><TR><TD ALIGN="center">Mexico</TD><TD ALIGN="center">0.482</TD><TD ALIGN="center">2012</TD></TR><TR><TD ALIGN="center">Chile</TD><TD ALIGN="center">0.503</TD><TD ALIGN="center">2011</TD></TR></TABLE><P>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. </P><P> </P>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com0tag:blogger.com,1999:blog-26706564.post-22258423834747468802015-02-02T16:19:00.000-05:002015-02-06T09:14:11.561-05:00A Cynical Take By Greece's Finance Minister On Mainstream Economists<P>I have found Yanis Varoufakis' 2014 book, <A HREF="http://www.routledgementalhealth.com/books/details/9781138923034/"><I>Economic Indeterminacy: A personnel encounter with economists' peculiar nemesis</I></A> 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 <A HREF="http://robertvienneau.blogspot.com/2014/12/first-formulation-of-folk-theorem-and.html">models</A> with <A HREF="http://robertvienneau.blogspot.com/2008/12/dont-say-there-must-be-something-common.html">multiple equilibria</A> and of <A HREF="http://robertvienneau.blogspot.com/2009/11/indeterminate-two-person-zero-sum-game.html">nondeterministic</A> outcomes. One might expect economists to <A HREF="http://robertvienneau.blogspot.com/2013/07/against-biotechnological-determinism.html">shy</A> away from these conclusions. </P><P>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. </P><P>But my point in this post is to quote from the preface: </P><BLOCKQUOTE><P>"...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.</P><P>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 <I>had</I> 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.)</P></BLOCKQUOTE><P>Varoufakis has some other books that sound interesting and more popular. I think his book; <A HREF="http://www.zedbooks.co.uk/paperback/the-global-minotaur"><I>The Global Minotaur: America, Europe and the Future of the Global Economy</I></A>; might be especially topical at the moment. </P><P><B>Update:</B> <A HREF="http://www.forbes.com/sites/stevekeen/2015/01/31/my-friend-yanis-the-greek-minister-of-finance/">Steve Keen</A> provides a link to one <A HREF="http://uadphil.econ.uoa.gr/UA/files/2013331662..pdf">exposition</A> of Varoufakis' argument that, in game theory, agents can and will deliberately choose irrational behavior. </P>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com0tag:blogger.com,1999:blog-26706564.post-74092975595906402412015-01-23T08:48:00.000-05:002015-01-23T08:48:00.164-05:00Approximating a Continuous Time Markov Process<TABLE CELLSPACING="1" CELLPADDING="1" BORDER="0" ALIGN="center"><TR><TD ALIGN="center"><a href="http://4.bp.blogspot.com/-qnp8oeUA9AI/VLUT7mXZ8YI/AAAAAAAAAeM/MQKZoNNDivE/s1600/Chain3State.jpg" imageanchor="1" ><img border="0" src="http://4.bp.blogspot.com/-qnp8oeUA9AI/VLUT7mXZ8YI/AAAAAAAAAeM/MQKZoNNDivE/s400/Chain3State.jpg" /></a></TD></TR><TR><TD ALIGN="center"><B>Figure 1: Rate of Transitions Between States in a Three-State Markov Chain</B></TD></TR></TABLE><B>1.0 Introduction</B><P>This post, about Markov processes, does not have much to do with <A HREF="http://robertvienneau.blogspot.com/2008/07/two-roads-diverged-in-yellow-wood-and.html">economics</A>. 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. </P><B>2.0 Continuous Time Markov Chains</B><P>Consider a stochastic process that, at any non-negative time <I>t</I> is in one of <I>N</I> states. Assume this process satisfies the Markov process: the future history of the process after time <I>t</I> depends only on the state of the process at time <I>t</I>, 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. </P><P>Formally, let <I>P</I><SUB><I>i</I>, <I>J</I></SUB> denote the conditional probability that the next transition will be into state <I>j</I>, given that the process is in state <I>i</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><BLOCKQUOTE><I>P</I><SUB><I>i</I>, <I>i</I></SUB> = 0; <I>i</I> = 0, 1, ..., <I>N</I> - 1 </BLOCKQUOTE><P>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: </P><BLOCKQUOTE><I>F</I><SUB><I>i</I>, <I>j</I></SUB>(<I>t</I>) = 1 - e<SUP>- λ<SUB><I>i</I></SUB> <I>t</I></SUP>; <I>i</I>, <I>j</I> = 0, 1, ..., <I>N</I> - 1; </BLOCKQUOTE><P>where <I>F</I><SUB><I>i</I>, <I>j</I></SUB>(<I>t</I>) is the conditional probability that the next transition will be before time <I>t</I>, given that the chain is in state <I>i</I> at time zero and that the next transition will be into state <I>j</I>. In other words, <I>F</I><SUB><I>i</I>, <I>j</I></SUB>(<I>t</I>) 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. </P><P>Let <I>P</I><SUB><I>i</I>, <I>j</I></SUB>(<I>t</I>) be the conditional probability that the chain is in state <I>j</I> at time <I>t</I>, given that the chain is in state <I>i</I> at time zero. These conditional probabilities satisfy Kolmogorov's forward equation: </P><BLOCKQUOTE><a href="http://4.bp.blogspot.com/-_y5SsLoi9cU/VLVkS5Ql9VI/AAAAAAAAAfQ/oucj897nExI/s1600/ForwardEquation0.jpg" imageanchor="1" ><img border="0" src="http://4.bp.blogspot.com/-_y5SsLoi9cU/VLVkS5Ql9VI/AAAAAAAAAfQ/oucj897nExI/s400/ForwardEquation0.jpg" /></a><BLOCKQUOTE><a href="http://2.bp.blogspot.com/-YdNK_g2gKNk/VLVkDVaJebI/AAAAAAAAAfI/bqXuF2z9So4/s1600/ForwardEquation1.jpg" imageanchor="1" ><img border="0" src="http://2.bp.blogspot.com/-YdNK_g2gKNk/VLVkDVaJebI/AAAAAAAAAfI/bqXuF2z9So4/s400/ForwardEquation1.jpg" /></a>, </BLOCKQUOTE></BLOCKQUOTE><P>where the transition rate matrix <B>Q</B> is defined to be: </P><BLOCKQUOTE><a href="http://4.bp.blogspot.com/-CtHzbgR_J9M/VLVjtBhEGRI/AAAAAAAAAfA/UmU5YcfpfUE/s1600/TransitionRateMatrix.jpg" imageanchor="1" ><img border="0" src="http://4.bp.blogspot.com/-CtHzbgR_J9M/VLVjtBhEGRI/AAAAAAAAAfA/UmU5YcfpfUE/s400/TransitionRateMatrix.jpg" /></a></BLOCKQUOTE><P>The elements in each row of the transition rate matrix sum to zero. Kolmogorov's forward equation can be expressed in scalar form: </P><BLOCKQUOTE><a href="http://4.bp.blogspot.com/-lLA57oKwLUU/VLVjXmWN7FI/AAAAAAAAAe4/gQzhgYugauE/s1600/ScalarEquation.jpg" imageanchor="1" ><img border="0" src="http://4.bp.blogspot.com/-lLA57oKwLUU/VLVjXmWN7FI/AAAAAAAAAe4/gQzhgYugauE/s400/ScalarEquation.jpg" /></a></BLOCKQUOTE><P>The above equation applies to continuous time Markov chains with a countably infinite number of states only under certain special conditions. </P><P>Steady state probabilities of this Markov chain satisfy: </P><BLOCKQUOTE><B>p</B> <B>Q</B> = <B>0</B>, </BLOCKQUOTE><P>where <B>p</B> is a row vector in which each element is the steady-state probability that the chain is in the corresponding state. </P><B>3.0 Discrete Time Approximation</B><P>A discrete time Markov chain is specified by a state transition matrix <B>A</B>, where <I>a</I><SUB><I>i</I>, <I>j</I></SUB> is the probability that the chain will transition in a time step from state <I>i</I> to state <I>j</I>, given that the chain is in state <I>i</I> at the start of the time step. Steady state probabilities for a discrete time Markov chain satisfy: </P><BLOCKQUOTE><B>p</B> <B>A</B> = <B>p</B></BLOCKQUOTE><P>The above equation compares and contrasts with how steady state probabilities relate to the transition rate matrix in a continuous time Markov chain. </P><P>Let the time step <I>h</I> 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>i</I>: </P><BLOCKQUOTE>P(0 or 1 transitions in time <I>h</I> | Chain in state <I>i</I> at time 0) = <BLOCKQUOTE>(1 + λ<SUB><I>i</I></SUB> <I>h</I>) e<SUP>- λ<SUB><I>i</I></SUB> <I>h</I></SUP></BLOCKQUOTE></BLOCKQUOTE><P>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: </P><BLOCKQUOTE><I>a</I><SUB><I>i</I>, <I>i</I></SUB> = e<SUP>- λ<SUB><I>i</I></SUB> <I>h</I></SUP> = e<SUP>- <I>q</I><SUB><I>i</I>, <I>i</I></SUB> <I>h</I></SUP>; <I>i</I> = 0, 1, ..., <I>N</I> - 1 </BLOCKQUOTE><P>The probability that the chain transitions to state <I>j</I>, given the chain is in state <I>i</I> at the start of the time step, is the product of: </P><UL><LI>The probability that a transition occurs during that time step, and</LI><LI>The conditional probability that the next transition will be into state <I>j</I>, given the chain is in state <I>i</I> at the start of the time step.</LI></UL><P>The following equation specifies this probability: </P><BLOCKQUOTE><I>a</I><SUB><I>i</I>, <I>j</I></SUB> = (1 - <I>a</I><SUB><I>i</I>, <I>i</I></SUB>)<I>P</I><SUB><I>i</I>, <I>j</I></SUB> = (1 - <I>a</I><SUB><I>i</I>, <I>i</I></SUB>) <I>q</I><SUB><I>i</I>, <I>j</I></SUB>/(- <I>q</I><SUB><I>i</I>, <I>i</I></SUB>); <I>i</I> ≠ <I>j</I></BLOCKQUOTE><P>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. </P><B>4.0 An Example</B><P>Consider a three-state, continuous time Markov chain. Figure 1 shows the rate of transitions between the various states. The transition rate matrix is: </P><BLOCKQUOTE><a href="http://2.bp.blogspot.com/-I5kOxMCM044/VLUWYqa7qqI/AAAAAAAAAeg/WXeGqTio1G4/s1600/ContinuousChain3State.jpg" imageanchor="1" ><img border="0" src="http://2.bp.blogspot.com/-I5kOxMCM044/VLUWYqa7qqI/AAAAAAAAAeg/WXeGqTio1G4/s400/ContinuousChain3State.jpg" /></a></BLOCKQUOTE><P>To discretize time, choose a small time step <I>h</I> such that, for all states <I>i</I>, the following probabilities are approximately unity: </P><BLOCKQUOTE>P(0 or 1 transitions in time <I>h</I> | Chain in state 0 at time 0) = <BLOCKQUOTE>[1 + (λ<SUB>0, 1</SUB> + λ<SUB>0, 2</SUB>)<I>h</I>] e<SUP>-(λ<SUB>0, 1</SUB> + λ<SUB>0, 2</SUB>)<I>h</I></SUP></BLOCKQUOTE></BLOCKQUOTE><BLOCKQUOTE>P(0 or 1 transitions in time <I>h</I> | Chain in state 1 at time 0) = <BLOCKQUOTE>[1 + (λ<SUB>1, 0</SUB> + λ<SUB>1, 2</SUB>)<I>h</I>] e<SUP>-(λ<SUB>1, 0</SUB> + λ<SUB>1, 2</SUB>)<I>h</I></SUP></BLOCKQUOTE></BLOCKQUOTE><BLOCKQUOTE>P(0 or 1 transitions in time <I>h</I> | Chain in state 2 at time 0) = <BLOCKQUOTE>[1 + (λ<SUB>2, 0</SUB> + λ<SUB>2, 1</SUB>)<I>h</I>] e<SUP>-(λ<SUB>2, 0</SUB> + λ<SUB>2, 1</SUB>)<I>h</I></SUP></BLOCKQUOTE></BLOCKQUOTE><P>The state transition matrix <B>A</B> for the discrete-time Markov chain is: </P><BLOCKQUOTE><a href="http://4.bp.blogspot.com/-ev83NCW-ycY/VLUWgDR5ksI/AAAAAAAAAeo/wnNK0Lb7v9U/s1600/DiscreteChain3State.jpg" imageanchor="1" ><img border="0" src="http://4.bp.blogspot.com/-ev83NCW-ycY/VLUWgDR5ksI/AAAAAAAAAeo/wnNK0Lb7v9U/s400/DiscreteChain3State.jpg" /></a></BLOCKQUOTE><P>I have not tested the above with concrete values for a continuous time Markov chain. </P><B>Reference</B><UL><LI>S. M. Ross (1970). <I>Applied Probability Models with Optimization Applications</I>. San Francisco: Holden-Day</LI></UL>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com0tag:blogger.com,1999:blog-26706564.post-72519912701404027102015-01-16T07:58:00.000-05:002015-01-16T07:58:00.237-05:00Laughing At Neoclassical Economists, Elsewhere<UL><LI>Matthew Yglesias <A HREF="http://www.vox.com/2015/1/13/7532671/9-things-only-neoclassical-economists-will-understand">lists</A> "Nine Things Only Neoclassical Economists Will Understand". Strangely, his twitter <A HREF="https://twitter.com/mattyglesias/status/555065312333553666">announcement</A> of this article is about a tenth.</LI><LI>Noah Smith <A HREF="http://www.bloombergview.com/articles/2015-01-15/nine-economics-mysteries-explained">purports</A> 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".</LI><LI>Tyler Cowen <A HREF="http://marginalrevolution.com/marginalrevolution/2015/01/i-liked-noah-smiths-piece-and-i-really-dont-mean-to-pick-on-him-but.html">attempts</A> to clarify the Heckscher-Ohlin theorem, but fails to note that "capital" <A HREF="http://robertvienneau.blogspot.com/2007/12/qualms-on-foreign-trade.html">cannot</A> be a factor of production in the Heckscher-Ohlin-Samuelson model. (He does note Leontief's empirical demonstration that the theory fails.)</LI></UL>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com0tag:blogger.com,1999:blog-26706564.post-57202198858455098512015-01-10T13:34:00.001-05:002015-01-16T06:25:07.163-05:00Because Something Is Happening Here/But You Don’t Know What It Is/Do You, Mister Jones?<P>Strangely, some prominent, somewhat liberal, economics bloggers have decided simultaneously to complain about (unnamed) left-leaning heterodox economists: </P><UL><LI>Noah Smith, at <A HREF="http://www.bloombergview.com/articles/2015-01-07/economics-stars-swing-left"><I>Bloomberg View</I></A> and at his <A HREF="http://noahpinionblog.blogspot.com/2015/01/sci-jacking.html">blog</A>.</LI><LI>Simon Wren-Lewis, at his <A HREF="http://mainlymacro.blogspot.com/2015/01/heterodox-laziness-or-worse.html">blog</A>.</LI><LI>Paul Krugman, at his <I>New York Times</I> <A HREF="http://krugman.blogs.nytimes.com/">blog</A>.</LI></UL><P>All three, incorrectly in my view, think the heterodox economists who they object to are only arguing politics. As far as I know, many, including me, do not take issue with Krugman's short-term policy views. Smith, in his trollish approach, raises a side comment about Austrian economists and the Mont Pelerin society. (I will state the proper label for Friedrich Hayek and Ludwig Von Mises is "economist", not "quasi-economist", as Smith would have it. But I've seen for some time that I am more well-informed on <A HREF="http://robertvienneau.blogspot.com/search/label/Austrian%20School%20Of%20Economics">Austrian economics</A> than Smith is.) </P><P>I think more pertinent issues center around modeling approaches, the image the profession projects in the public sphere, and the sociology of the profession. How is it than so many rightists have been able to <A HREF="http://robertvienneau.blogspot.com/search/label/Making%20Life%20More%20Brutish">push</A> the view that their politics is good economics, while simultaneously insisting that economics is a positive science? The involvement of economists with neoliberal politics is not confined to some fringe. Consider, for example, the Chicago school, the lack of a strong ethics policy in the American Economic Association, the Washington consensus, and even Paul Samuelson's 1960s research that led to to Efficient Market Hypothesis. </P><P>There is probably also a personnel element here. Non-mainstream, heterodox economists would like more acknowledgement by mainstream economists. If your knowledge of heterodox economics is confined to what you can get off the Internet, aside from what professional literature is now available there, you might not know what you are talking about when you talk about heterodox economics. (And this includes the Austrian school.) Furthermore, when you develop parallel ideas, or draw on heterodox economics, you should acknowledge it. In the linked post above, Krugman makes the point that "a country that borrows in its own currency" cannot easily become like Greece, under attack from "bond vigilantes", without saying anything about <A HREF="http://robertvienneau.blogspot.com/search/label/Endogenous%20Money">endogenous money</A> or the economists at the University of Missouri Kansas City. (I could also say something about the research for which Krugman won the "Nobel Prize".) If you know where to look, you can find Joseph Stiglitz acknowledging that he learned a lot from such Cambridge economists as Nicky Kaldor and Joan Robinson. </P><P>Maybe economics would be a better place if the center of gravity in economics in the United States were arguments between mainstream economists and, say, economists at the New School and the University of Massachusetts at Amherst. If the profession were to move in this direction, young doctorates would need to be socialized to not dismiss viewpoints because of the rankings of the universities and the journals in which they were advanced. Methodology would continue to need to be broadened to include more than mathematical models of optimizing agents. </P><P><B>Update:</B> Reactions from <A HREF="http://stumblingandmumbling.typepad.com/stumbling_and_mumbling/2015/01/heterodox-economics-the-left-.html">Chris Dillow</A>, <A REF="http://econospeak.blogspot.com/2015/01/to-kick-off-2015-hippie-punching.html">Peter Dorman</A>, and <A HREF="http://www.alexsarchives.org/2015/01/economists-and-politics/">Alex Marsh</A>. </P>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com2tag:blogger.com,1999:blog-26706564.post-76681919848673274642015-01-09T07:47:00.000-05:002015-01-09T07:47:14.724-05:00Greg Mankiw, Fool Or Knave?<P>Greg Mankiw seems determined to continually attempt to bring <A HREF="http://robertvienneau.blogspot.com/2013/09/seth-ackerman-and-mike-beggs-on.html">his</A> <A HREF="http://robertvienneau.blogspot.com/2008/07/greg-mankiw-still-fool.html">supposed</A> <A HREF="http://robertvienneau.blogspot.com/2007/11/greg-mankiw-ignorant-or-dishonest.html">profession</A> <A HREF="http://robertvienneau.blogspot.com/2007/07/dont-believe-word-greg-mankiw-says.html">into</A> <A HREF="http://robertvienneau.blogspot.com/2007/05/greg-mankiw-incompetent-or-dishonest.html">disrepute</A>. Last week, at the annual meeting of American economists (the Allied Social Science Associations), Greg Mankiw <A HREF="https://larspsyll.wordpress.com/2015/01/06/the-mankiw-piketty-showdown-at-the-assa-annual-meeting-january-2015/">chaired</A> a session on Thomas Piketty's <I>Capital in the 21st Century</I>. In his draft of his prepared remarks, Mankiw writes: </P><BLOCKQUOTE>"Equation (3) says that capital earns its marginal product." -- Greg Mankiw, "<A HREF="https://www.aeaweb.org/aea/2015conference/program/retrieve.php?pdfid=520">Yes, <I>r</I> > <I>g</I>. So what?</A>" (24 November 2014). </BLOCKQUOTE><P>Because of price Wicksell effects, the marginal product of finance capital is generally unequal, in equilibrium, to the rate of profits. Even Champernowne's <A HREF="http://robertvienneau.blogspot.com/2009/05/neoclassical-response-to-cambridge.html">chain index</A>, which abstracts from price Wicksell effects, cannot generally be used to defend the equality in aggregate models of the rate of profits and the marginal product of capital. Economic theory imposes no restriction on the direction of price and <A HREF="http://robertvienneau.blogspot.com/2009/11/taxonomy-of-effects-of-wicksell-effects.html">real Wicksell effects</A>, and the chain index is not well-defined in the presence of positive Wicksell effects. Neoclassical theory claims, at best, that the price of each capital good is equal, in equilibrium, to its marginal product. But marginal productivity is <I>not</I> a theory of the functional income distribution, since every point on the wage-rate of profits frontier <A HREF="http://robertvienneau.blogspot.com/2013/04/choice-of-technique-with-smooth.html">is</A> <A HREF="http://robertvienneau.blogspot.com/2013/04/choice-of-technique-two-good-model-cobb.html">compatible</A> with all valid marginal productivity conditions. Even if the returns to capital could be explained by marginal productivity, this would not justify any particular size of the <A HREF="http://robertvienneau.blogspot.com/2009/03/business-as-sabotage.html">tolls</A> that capitalists are able to impose. A conceptual <A HREF="http://robertvienneau.blogspot.com/2012/01/nells-diagram-of-capitalist-economy.html">distinction</A> can and should be made between the cost of capital goods and the returns to capitalists. </P><P>As far as I am concerned, the above is just good economics, agreed to by all non-ignorant economists, neoclassical or otherwise. But the confusion and general muddleheadness promoted by such as Mankiw, seems to serve a functional <A HREF="http://robertvienneau.blogspot.com/2007/01/institutionalist-before-joan-robinson.html">purpose</A> in the sublunary world. </P>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com0tag:blogger.com,1999:blog-26706564.post-3138307514534463742014-12-29T10:36:00.000-05:002014-12-29T10:36:29.750-05:00On "Privatized Keynesianism"<P>I have been reading Colin Crouch's <A HREF="http://www.amazon.com/Strange-Non-death-Neo-liberalism-Colin-Crouch/dp/0745652212"><I>The Strange Non-Death of Neoliberalism</I></A><SUP>1</SUP>. A major theme is that an ideological divide between more reliance on markets and on government misses issues raised by the existence of large - including multinational - corporations. The <A HREF="http://robertvienneau.blogspot.com/2014/11/for-conflating-neoliberalism-and.html">neoliberal</A> assault on government has been increasing the strength of corporations, not competitive markets. Furthermore, corporations have been <A HREF="http://robertvienneau.blogspot.com/2011/10/corporate-governance.html">taking</A> on the <A HREF="http://robertvienneau.blogspot.com/2010/05/performing-corporate-finance.html">role</A> of <A HREF="http://robertvienneau.blogspot.com/2013/03/governance-as-used-in-current.html">government</A>. Crouch mentions, for example, the "seconding" of corporate executives to various ministries; the likelihood that internal policies of a Multi-National Corporation on, say, child labor may be more restrictive than laws in many third world countries; and the role of corporations in <A HREF="https://www.oasis-open.org/standards">setting</A> <A HREF="http://www.itu.int/en/ITU-T/Pages/default.aspx">international</A> <A HREF="https://www.commoncriteriaportal.org/">standards</A>, where organizations with nation-states may be weak.</P><P>But my point in this post is to note Crouch's introduction(?) of a new technical term, <I>Privatized Keynesianism</I>. A contrast between the post-World War II golden age and the later neoliberal era<SUP>2</SUP> is needed to make sense of this term. After the war, in the United States - and, I gather, in other advanced industrial capitalist economies - wages rose with average productivity. Furthermore, governments, under a <A HREF="http://en.wikipedia.org/wiki/Employment_Act_of_1946">somewhat</A> Keynesian ideology, saw it as their responsibility to maintain aggregate demand. These conventions came undone in the 1970s. Productivity increased (at a slower pace), but wages failed to keep up, and governments came to emphasize fighting inflation, not unemployment.</P><P><A HREF="http://robertvienneau.blogspot.com/2010/08/why-income-equality-leads-to.html">Increased inequality</A>, however, did not eliminate the need to manage aggregate demand. Neither consumer spending from wages nor an abdication from fiscal polity by government could fill this lacuna. This period saw the increased availability of debt, the creation of secondary markets for the trading of bets on bets on bundles of debts (derivatives), and the capture of credit rating agencies by sellers of debts. This institutional structure led to the collective, but private, macroeconomic regulation of aggregate demand<SUP>3</SUP>. This institutional structure is what Crouch calls privatized Keynesianism<SUP>4</SUP>. The irresponsibility of banks, in some sense, produced a (temporary, unsustainable) positive externality.</P><B>Footnotes</B><OL><LI>I might as well note two mistakes I found irritating. Somewhere in one of the early chapters, Crouch, who I gather is British, refers to Eugene McCarthy when he means Joe McCarthy. I also thought that Crouch's account of the role of Fanny Mae and Freddy Mac in subprime mortages reflected too much credence for right-wing liars.</LI><LI>I date the start of the neoliberal era with Nixon ending the fixed exchange rate between the United States dollar and gold, a major element of the Bretton Woods system.</LI><LI>Is this a non-microfounded, functionalist account?</LI><LI>From this perspective, the accumulation of private debt was a symptom, not the ultimate cause of the recent Global Financial Crisis, a cause that has yet to be addressed. These ideas seem to me to be close to Thomas Palley's <A HREF="http://www.amazon.com/Financial-Crisis-Stagnation-Destruction-Prosperity/dp/1107612462">Structural Keynesianism</A>. Has anybody read James K. Galbraith's <A HREF="http://books.simonandschuster.com/The-End-of-Normal/James-K-Galbraith/9781451644920"><I>The End of Normal: The Great Crisis and the Future of Growth</I></A>?</LI></OL>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com8tag:blogger.com,1999:blog-26706564.post-617011939287526442014-12-18T14:13:00.001-05:002014-12-18T14:45:23.551-05:00Slaves Identifying With Their Masters<P>Marx's attempt to describe how capitalism <A HREF="http://robertvienneau.blogspot.com/2008/07/marx-was-skint-but-he-had-sense-engels.html">creates</A> <A HREF="http://robertvienneau.blogspot.com/2014/04/illusions-generated-by-markets-like.html">objective</A> <A HREF="http://robertvienneau.blogspot.com/2013/08/preliminary-thoughts-on-volume-two-of.html">illusions</A>, so to speak, is one aspect of <I>Capital</I> that I like. In <A HREF="http://crookedtimber.org/2009/04/20/explaining-marx-to-newbies/#comment-273465">this</A> comment on a long-ago Crooked Timber <A HREF="http://crookedtimber.org/2009/04/20/explaining-marx-to-newbies/">post</A>, "Ted" draws an analogy to J. S. Mill's <I>Subjection of Women</I>, which I have never read. Apparently, Mill explains how women can come to identify with their oppressors. </P><P>I happen to currently be reading the autobiography of local Rochester <A HREF="http://www.fdrc-rochester.com/">hero</A>, Frederick Douglass. This passage identifies a curious phenomenon: </P><BLOCKQUOTE>"Moreover, slaves are like other people, and imbibe prejudices quite common to others. They think their own better than that of others. Many, under the influence of this prejudice, think their own masters are better than the masters of other slaves; and this, too, in some cases, when the very reverse is true. Indeed, it is not uncommon for slaves even to fall out and quarrel among themselves about the relative goodness of their masters, each contending for the superior goodness of his own over that of the others. At the very same time, they mutually execrate their masters when viewed separately. It was so on our plantation. When Colonel Lloyd's slaves met the slaves of Jacob Jepson, they seldom parted without a quarrel about their masters; Colonel Lloyd's slaves contending that he was the richest, and Mr. Jepson's slaves that he was the smartest, and most of a man. Colonel Lloyd's slaves would boast his ability to buy and sell Jacob Jepson. Mr. Jepson's slaves would boast his ability to whip Colonel Lloyd. These quarrels would almost always end in a fight between the parties, and those that whipped were supposed to have gained the point at issue. They seemed to think that the greatness of their masters was transferable to themselves. It was considered as being bad enough to be a slave; but to be a poor man's slave was deemed a disgrace indeed." -- Frederick Douglas, <I>Narrative of the Life of Frederick Douglas</I></BLOCKQUOTE><P>Maybe some day I'll read Hegel's <I>Phenomenology of Spirit</I> - it is on my shelf - to learn some ideas about the master-slave dialetic. Mayhaps the above is analogous to the opinions of many <A HREF="http://robertvienneau.blogspot.com/2006/06/capitalism-seen-as-unfree-in-its.html">wage</A>-<A HREF="http://robertvienneau.blogspot.com/2010/08/when-adam-delved-and-eve-span-who-was.html">slaves</A>. There seem to be many ways to be unfree, and many ways to deny this. </P>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com4tag:blogger.com,1999:blog-26706564.post-36390946154693549462014-12-12T11:37:00.000-05:002014-12-12T11:37:09.588-05:00First Formulation of Folk Theorem and Indeterminacy in Game Theory<TABLE CELLSPACING="1" CELLPADDING="1" BORDER="0" ALIGN="center"><TR><TD ALIGN="center"><a href="http://4.bp.blogspot.com/-Fh7gKDYINAk/VIsZBXMS9uI/AAAAAAAAAd8/M4Ed8nFItNM/s1600/RockPaperScissorsChaos.jpg" imageanchor="1" ><img border="0" src="http://4.bp.blogspot.com/-Fh7gKDYINAk/VIsZBXMS9uI/AAAAAAAAAd8/M4Ed8nFItNM/s320/RockPaperScissorsChaos.jpg" /></a></TD></TR><TR><TD ALIGN="center"><B>Initial and Chaotic Learning in Rock-Paper-Scissors</B></TD></TR></TABLE><P>Consider a game, as games are defined in game theory. And consider some strategy for some player in some game. The <I>folk theorem</I> states, roughly, that any strategy can be justified as a solution for a game by considering an infinitely repeated game. (An amusing corollary might be stated as saying that competition is the same as monopoly, if you do the <A HREF="http://arxiv.org/abs/1201.3798">math</A> <A HREF="http://robertvienneau.blogspot.com/2013/04/perfect-competition-is-same-as-monopoly.html">right</A>.) The following seems to me to state the folk theorem (abstracting from the distinction between Nash equilibria and Von Neumann and Morgenstern's solution concept): </P><BLOCKQUOTE><P>"<B>21.2.3.</B> If our theory were applied as a statistical analysis of a long series of plays of the same game - and not as the analysis of one isolated play - an alternative interpretation would suggest itself. We should then view agreements and all forms of cooperation as establishing themselves by repetition in such a long series of plays.</P><P>It would not be impossible to derive a mechanism of enforcement from the player's desire to maintain his record and to be able to rely on the on the record of his partner. However, we prefer to view our theory as applying to an individual play. But these considerations, nevertheless, possess a certain signiificance in a virtual sense. The situation is similar to the one we encountered in the analysis of the (mixed) strategies of a zero-sum two-person game. The reader should apply the discussions of 17.3 <I>mutatis mutandis</I> to the present situation." -- John Von Neumann and Oscar Morgenstern (1953) p. 254.</P></BLOCKQUOTE><P>I have heard it claimed that economic theory has developed such that any moderately informed graduate student can now provide you with a model that yields any conclusion that you like. The folk theorem, as I understand it, is not even the most threatening finding for the ability of game theory to yield determinate conclusions.</P><P>Consider an iterated game before an equilibrium, under some definition or another, has been achieved. The players are trying to learn each others' strategies. Even a simple game, such as Rock-Scissors-Paper, can yield chaotic dynamics (Sato, Akiyama, and Farmer 2002; Galla and Farmer 2013). An equilibrium might never be established, for it is worthwhile for some players to deliberately choose "irrational" moves so as to ensure that other players do not achieve equilibrium, instead of a result that benefits the supposedly irrational player (Foster and Young 2012). (I hope I found this reference from reading Yanis Varoufakis, who, in one paper in one of his books, makes this point with the centipede game.) Apparently, this irrationality does not disappear by moving towards a more meta-theoretic level. And one player, who understands the evolutionary behavior of the other player in a Prisoner's Dilemma, can manipulate the other player to result in a asymmetric result - that is, a case where the non-evolutionary player extorts the player following a mindless evolutionary strategy (Press and Dyson 2012, Stewart and Plotkin 2012). </P><B>References</B><UL><LI>Dean P. Foster and H. Peyton Young (2001). <A HREF="http://www.pnas.org/content/98/22/12848.abstract">On the impossibility of predicting the behavior of rational agents</A>. <I>Proceedings of the National Academy of Sciences</I>. V. 98: pp. 12848-12853.</LI><LI>Tobias Galla and J. Doyne Farmer (2013). <A HREF="http://www.pnas.org/content/110/4/1232.abstract">Complex dynamics in learning complicated games</A>. <I>Proceedings of the National Academy of Sciences</I>, V. 110: pp. 1232-1236.</LI><LI>Tiago P. Peixoto and Stefan Bornholdt (2013). <A HREF="http://arxiv.org/abs/1201.3798">No need for conspiracy: Self-organized cartel formation in a modified trust game</A>.</LI><LI>William H. Press and Freeman J. Dyson (2012). <HREF="http://www.pnas.org/content/109/26/10409.abstract">Iterated Prisoner's Dilemma contains strategies that dominate any evolutionary opponent</A>. <I>Proceedings of the National Academy of Sciences</I>.</LI><LI>Yuzuru Sato, Eizo Akiyama, and J. Doyne Farmer (2002). <A HREF="http://www.pnas.org/content/99/7/4748.abstract">Chaos in learning a simple two-person game</A>. <I>Proceedings of the National Academy of Sciences</I>. V. 99: pp. 4748-4751.</LI><LI>Alexander J. Stewart and Joshua B. Plotkin (2012). <HREF="http://www.pnas.org/content/110/38/15348.abstract">From extortion to generosity, evolution in the iterated Prisoner's Dilemma</A>. <I>Proceedings of the National Academy of Sciences</I>.</LI><LI>Yanis Varoufakis (2005). Rational Rules of Thumb in Finite Dynamic Games: N-person Backward Induction with Inconsistenly Aligned Beliefs and Full Rationality. <I>American Journal of Applied Sciences</I>.</LI><LI>Yanis Varoufakis (2013). <A HREF="http://www.amazon.com/Economic-Indeterminacy-encounter-economists-Routledge/dp/0415668492/"><I>Economic Indeterminacy: A personal encounter with the economists' peculiar nemesis</I></A>.</LI><LI>John Von Neumann and Oscar Morgenstern (1953). <I>Theory of Games and Economic Behavior</I>, third edition. Princeton University Press.</LI></UL>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com0tag:blogger.com,1999:blog-26706564.post-1328452965092042562014-12-03T09:36:00.000-05:002015-01-12T06:42:32.094-05:00Noah Smith Befuddling Bloomberg Readers<B>1.0 Introduction</B><P>Noah Smith seems to be trying to become a professional columnist and blogger, however his day job works out. I do not know if the same opportunity still exists, as it apparently did when, for example, <A HREF="http://www.eschatonblog.com/">Duncan Black</A>, <A HREF="http://www.motherjones.com/authors/kevin-drum">Kevin Drum</A>, <A HREF="http://www.vox.com/authors/ezra-klein">Ezra Klein</A>, <A HREF="http://talkingpointsmemo.com/">Josh Marshall</A>, <A HREF="http://www.salon.com/writer/heather_digby_parton/">Heather Parton</A>, and <A HREF="http://www.vox.com/authors/matthew-yglesias">Matthew Yglesias</A> were starting out. I do not want to spend much <A HREF="http://robertvienneau.blogspot.com/2014/09/hayek-not-opposed-to-keynes-on.html">time</A> <A HREF="http://robertvienneau.blogspot.com/2014/05/need-for-engagement-with-heterodox.html">taking</A> <A HREF="http://crookedtimber.org/2014/12/02/economists-arent-superior-just-because/">down</A> <A HREF="http://crookedtimber.org/2012/12/27/noah-smith-had-me-going-for-a-minute-there/">Smith</A>, but I wish so many of his columns did not provide anecdotal evidence that the job of mainstream economists is to sow confusion into the public sphere. Maybe I should try to resolve not to read him. </P><B>2.0 Confusion on Marginal Productivity</B><P>Consider this <I>Bloomberg</I> <A HREF="http://www.bloombergview.com/articles/2014-11-28/you-want-a-bigger-paycheck-convince-me">column</A>, "You want a bigger paycheck? Convince me." Smith's column contains the, I guess, still obligatory confused red-baiting: </P><BLOCKQUOTE>"No economic model says that people get paid based on average productivity. If they did, there would be no income left over for capital -- no profits, rents or interest. We’d be living in a sort of a Marxist world, where labor is the only thing with any value." -- Noah Smith </BLOCKQUOTE><P>I do not see what that comment has to do with Marxism. (Consider the <I>Critique of the Gotha Program</I>.) Anyways, this comment immediately follows Smith's graphical and empirical demonstration that real wages rose with increases in productivity in the United States during the post war golden age. Was the United States in the 1950s and 1960s a "sort of Marxist" society? Certainly economic models of <A HREF="http://robertvienneau.blogspot.com/2010/08/why-income-equality-leads-to.html">growth</A> <A HREF="http://robertvienneau.blogspot.com/2009/02/simple-and-expanded-reproduction.html">and</A> <A HREF="http://robertvienneau.blogspot.com/2014/10/r-g-in-steady-state.html">distribution</A> exist for thinking about the relationship between wages and average productivity in the golden age, and the breakdown of this relationship in the subsequent neoliberal era. </P>Smith apparently thinks that the theory of marginal productivity is a theory of the distribution of income. He is, <A HREF="http://robertvienneau.blogspot.com/2013/04/choice-of-technique-with-smooth.html">of</A> <A HREF="http://robertvienneau.blogspot.com/2013/04/choice-of-technique-two-good-model-cobb.html">course</A>, <A HREF="http://robertvienneau.blogspot.com/2007/05/greg-mankiw-incompetent-or-dishonest.html">quite</A> <A HREF="http://robertvienneau.blogspot.com/2006/09/non-teaching-of-empirical-superiority.html">mistaken</A>. Even worse, Smith goes on to use the discredited Solovian growth model, with an aggregate Cobb-Douglas production function, to explain how economists supposedly explain (changes in) the shares of "capital" and labor in national income. </P><P>Is it progress that Smith does not bring up skills-biased technical change, a nonsensical theory often used to propagandize for increased inequality in the distribution of wages? Maybe not, for Smith's purpose seems to be to propagandize for increased inequality in the functional income distribution between "capital" and wages. And so he brings up an equally nonsensical theory about the "rise of robots". </P><B>3.0 Inadequate Understanding on Women in Economics</B><P>Even when I don't necessarily disagree with Smith, I often find his columns insufficiently informed. <A HREF="http://www.bloombergview.com/articles/2014-11-21/economics-is-a-dismal-science-for-women">Here</A> he writes about career prospects in economics for women. I thank Smith for bringing this <A HREF="http://www.psychologicalscience.org/pdf/Women-Academic-Science.pdf">paper</A> by Ceci, Ginther, Kahn, and Williams to my attention. But it takes <A HREF="https://ello.co/claudiasahm/post/XGMhy3oKQBMa5JVudwP82A">Claudia Sahm</A>, in a response to this column, to bring up the Committee on the Status of Women in the Economics Profession (CSWEP). And, as far as I am aware, nobody previously commenting on Smith has mentioned the <A HREF="http://www.iaffe.org/">International Association for Feminist Economics</A> (IAFFE) and their journal, <A HREF="http://www.feministeconomics.org/"><I>Feminist Economics</I></A>. If you want to <A HREF="http://worthwhile.typepad.com/worthwhile_canadian_initi/2014/11/is-economics-really-a-dismal-science-for-women.html?cid=6a00d83451688169e201b7c70e1cff970b#comment-6a00d83451688169e201b7c70e1cff970b">argue</A> that homo economicus is gendered, I suggest browsing back issues of that journal. </P>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com1tag:blogger.com,1999:blog-26706564.post-45785762921243662502014-11-21T14:53:00.002-05:002014-11-21T14:54:58.874-05:00Humans And Other Animals<TABLE CELLSPACING="1" CELLPADDING="1" BORDER="0" ALIGN="center"><TR><TD ALIGN="center"><iframe title="YouTube video player" width="320" height="260" src="http://www.youtube.com/embed/lKhAd0Tyny0" frameborder="0" allowfullscreen></iframe ></TD></TR><TR><TD ALIGN="center"><B>Figure 1: Chapuchin Monkeys, Our Cousins</B></TD></TR></TABLE><P>What do we think about generalizations, validated partly with experiments with non-human animals, for economics? </P><P><A HREF="http://robertvienneau.blogspot.com/2009/11/nicholas-georgescu-roegen.html">Nicholas Georgescu-Roegen</A> is an economist widely admired by heterodox economists. He quit the American Economic Association in response to their flagship publication, the <I>American Economic Review</I>, publishing articles on, if I recall correctly, pigeons. Researchers were trying to demonstrate that properly trained pigeons had downward-sloping demand curves. I gather they wanted to show income effects and substitution effects, as well, with these laboratory experiments. </P><P>On the other hand, are we not supportive of behavioral economists undermining utility theory? I am thinking of controlled experiments that demonstrate people do not conform to the axioms of preference theory. And some of these experiments, as illustrated in the YouTube video linked above, extend beyond humans. </P><P>I have a suggestion to resolve such a tension. One might want to treat investigations of humans as a naturalistic enterprise. If so, one would not want to impose an a priori boundary on the different constituents of minds. Whether some species of animals has some sense of self, expectations of the future, primitive languages, or what not should be found by empirical investigation. On the other hand, activities that depend on the existence of social institutions cannot be expected to be found in animals not embedded in any society. And demand curves, if they were to exist, would only arise in specific market institutions. </P><B>Reference</B><UL><LI>Philip Mirowski (1994). The realms of the Natural, in <I>Natural Images in Economic Thought</I> (ed. by P. Mirowski), Cambridge University Press.</LI></UL>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com0tag:blogger.com,1999:blog-26706564.post-20886420101927564702014-11-06T08:21:00.000-05:002014-11-06T08:21:00.471-05:00Income Distribution And A Simple Labor Theory Of Value<P>I have a new <A HREF="http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2519466">paper</A> available on the Social Science Research Network: </P><P><B>Title:</B> Income Distribution And A Simple Labor Theory Of Value: Empirical Results From Comprehensive International Data </P><P><B>Abstract:</B> This paper presents the results of an empirical exploration, with data from countries worldwide, of Sraffian, Marxian, and classical political economy. Income distribution, as associated with systems of prices of production, fails to describe many economies. Economies in most countries or regions lie near their wage-rate of profits frontier, when the frontier is drawn with a numeraire in proportions of observed final demands. Labor values predict market prices better than prices of production do. Labor values also predict market prices better than they predict prices of production. In short, a simple labor theory of value is a surprisingly accurate price theory for economies around the world. </P>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com3tag:blogger.com,1999:blog-26706564.post-55507909384162813922014-11-01T12:43:00.002-04:002014-11-04T10:01:47.172-05:00For Conflating Neoliberalism And Neoclassical Economics<P><A HREF="http://robertvienneau.blogspot.com/2009/07/principles-of-neoliberalism.html">Neoliberalism</A> is a political project to remake the world into an unrealizable utopia. Neoclassical economics is a supposedly scientific effort to <A HREF="http://robertvienneau.blogspot.com/2014/02/post-keynesianism-contrasted-with.html">explain</A> the world by its deviations from an unrealizable utopia. And they are both about how the world deviates from that utopia. This post is about this resemblance, not the differences, between neoliberalism and neoclassical economics. </P><P>This utopia consists of a society organized around markets<SUP>1</SUP>. These markets require government to define property rights and enforce contract law. But, in the utopia, they are not to be <I>embedded</I> in a broader institutional setting that prevents their supposedly free adjustment. Examples of government-imposed inference with such self-regulation include minimum wages, rent control, laws against price-gouging, usury laws, subsidies for farmers to limit the size of harvests so as to maintain their income, payments to the able-bodied unemployed<SUP>2</SUP>, and so on. Polanyi's claim is that such so-called interventions are bound to arise. The ideal which those enacting such laws were reacting against is <A HREF="http://robertvienneau.blogspot.com/2014/07/against-mixed-economy-1.html">unachievable</A>, anyways. In the ideal, land, labor, and capital are treated as if they are only commodities. But land is the natural setting in which the economy takes place, and labor and capital involve social relations that cannot be reduced only to market relationships. </P><P>Both neoliberals and neoclassical economists often recognize their utopia must be constructed<SUP>3</SUP>, that it, will not emerge <A HREF="http://robertvienneau.blogspot.com/2006/05/against-reification-of-property-rights_19.html">naturally</A>, in some sense. The solution for problems with markets is said to be to construct more markets. I think about the tragedy of the commons, the theory of externalities<SUP>4, 5</SUP>, and the emphasis in neoclassical welfare theory on Pareto optimality. A paradigmatic policy recommendation, for both neoliberals and neoclassical economics, is the establishment of markets for pollution permits. </P><B>Footnotes</B><OL><LI>I have been reading Block and Somers (2014), and I read Polanyi (1944) more than a decade ago.</LI><LI>Block and Somers approvingly cite revisionist history from Mark Blaug in the 1960s that challenged centuries-long interpretations of English Poor Laws, especially the Speedhamland system. I know Blaug through his (multi-edition) history of economics and his misrepresentations of Sraffians and the Cambridge Capital Controversy. So I was glad to see a cite where he seems to be correct.</LI><LI>This emphasis on the need for government to construct markets, to my mind, is a distinctive difference between classical liberals and sophisticated neoliberals.</LI><LI>Some mainstream economists defend themselves from critics by asserting that the critics attack a strawperson. Economists do not believe, they say, that markets are perfect. And they'll ask why are the critics not aware of the frequent teaching about externalities. This objection seems to me to be beside the point if neoclassical economists react, as many do, the existence of an externality by calling for policy for internalizing the externality (or, at least, imitating the result of such policies).</LI><LI>If one accepted neoclassical economics as a positive science, how could one call for any policy conclusion without an explicit statement of normative values at some low level of abstration?</LI></OL><B>References</B><UL><LI>Fred Block and Margaret R. Somers (2014). <I>The Power of Market Fundamentalism: Karl Polanyi's Critique</I>Harvard University Press.</LI><LI>Karl Polanyi (1944). <I>The Great Transformation: The Political and Economic Origins of Our Time</I>.</LI></UL>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com0tag:blogger.com,1999:blog-26706564.post-5597247313055534132014-10-31T08:20:00.000-04:002014-10-31T08:20:00.439-04:00Fred Lee<P><A HREF="http://econospeak.blogspot.com/2014/10/the-passing-of-fred-lee-another-old.html">Barkley Rosser</A>, <A HREF="http://anticap.wordpress.com/2014/10/24/fred-lee-rip/">David Ruccio</A>, and <A HREF="http://nakedkeynesianism.blogspot.com/2014/10/frederic-s-lee.html">Matias Vernengo</A> have obituaries. I find I had not known much about Lee's life. </P><P>I have been influenced by Lee's work on markup pricing (also known as full-cost pricing), the <A HREF="http://robertvienneau.blogspot.com/2010/02/fred-lees-new-history-of-heterodox.html">history</A> of <A HREF="http://robertvienneau.blogspot.com/2006/09/where-do-correct-ideas-come-from-do.html">heterodox</A> economics, and the suppression of heterodox economics by the mainstream through <A HREF="http://robertvienneau.blogspot.com/2010/05/alternative-economics-in-mechanics.html">bullying</A> <A HREF="http://robertvienneau.blogspot.com/2014/08/political-intervention-in-faculty.html">and</A> <A HREF="http://robertvienneau.blogspot.com/2011/02/letter-to-editor.html">bureaucratic</A> <A HREF="http://robertvienneau.blogspot.com/2007/03/neoclassicalism-as-lysenkoism-in-u-k.html">measures</A>. I think highly of Lee's 2004 paper (written in collaboration with Steve Keen), "The Incoherent Emperor: A Heterodox Critique of Neoclassical Microeconomic Theory". I can only find one blog <A HREF="http://robertvienneau.blogspot.com/2012/08/not-necessarily-i-could-be-arguing-in.html">post</A> of mine referencing this paper. Lee promoted <A HREF="http://robertvienneau.blogspot.com/2011/09/international-journal-of-pluralism-and.html">pluralism</A> in economics. </P>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com0tag:blogger.com,1999:blog-26706564.post-40671362059132185862014-10-24T16:39:00.000-04:002014-12-01T10:33:38.536-05:00Marginal Productivity Theory of Distribution: Acknowledged Blatherskite<P>I was surprised at how many reviews of Thomas Piketty's <I>Capital in the 21st Century</I> draw on the Cambridge Capital Controversy to argue that Piketty's theoretical framework is grossly inadequate. </P><UL><LI>James K. Galbraith's Spring 2014 <A HREF="http://www.dissentmagazine.org/article/kapital-for-the-twenty-first-century">review</A> in <I>Dissent</I>.</LI><LI>Dean Baker <A HREF="http://www.cepr.net/index.php/blogs/beat-the-press/the-ghost-of-joan-robinson-and-thomas-piketty">last</A> May.</LI><LI>October 2014 <A HREF="http://www.paecon.net/PAEReview/issue69/whole69.pdf">special issue</A> of the <I>Real-World Economics Review</I>.</LI><LI>Tony Aspromourgos's <A HREF="http://www.centrosraffa.org/public/bbc90001-bb13-4313-955c-fd9e0ea2df62.pdf">Thomas Piketty, the future of capitalism and the theory of distribution: a review essay</A> (October 2014) (<B>H/T:</B> <A HREF="http://nakedkeynesianism.blogspot.com/2014/10/tony-aspromourgos-on-thomas-piketty.html">David Fields</A>).</LI><LI>Javier Lopez Bernardo, Felix Lopez Martinez, and Engelbert Stockhammer's <A HREF="http://www.postkeynesian.net/downloads/wpaper/PKWP1411.pdf">A Post-Keynesian Response to Piketty's 'Fundamental Contradiction of Capitalism'</A> (October 2014) (<B>H/T</B> <A HREF="http://larspsyll.wordpress.com/2014/10/25/a-post-keynesian-response-to-piketty/">Lars P. Syll</A>). This response brings up, in addition to the CCC, the <A HREF="http://robertvienneau.blogspot.com/2014/10/r-g-in-steady-state.html">Post Keynesian theory of distribution</A>.</LI><LI>John Bellamy Foster and Michael D. Yates' <A HREF="http://monthlyreview.org/2014/11/01/piketty-and-the-crisis-of-neoclassical-economics/">review</A></LI> in <I>Monthly Review</I></UL><P>I like this Aspromourgos quote: </P><BLOCKQUOTE>However classical the questions Piketty addresses, when he turns to explain the determination of r he has recourse to the conventional, post-classical marginal productivity theory of distribution: diminishing marginal capital productivity is 'natural' and 'obvious' (212–16). (He is much less willing to have recourse to time preference: 358–61; <I>cf</I>. 399–400.) The logical critique of capital aggregates – applied either at the macro or micro level – as supposed independent explanatory variables in the theory of profit rates, first coherently stated by Piero Sraffa (1960, pp. 81–7; see also Kurz and Salvadori 1995, pp. 427–67), is nowhere acknowledged or addressed. That such a relatively well-read economist as Piketty can so unhesitatingly apply this bankrupt approach, is testament to how completely a valid body of critical theoretical analysis can be submerged and forgotten in social science (a phenomenon for the sociologists of knowledge to contemplate). This is so, notwithstanding that Piketty offers a brief interpretation of the 'Cambridge' capital debates, making them turn upon the issues of whether there is substitutability in production (and associated flexibility of capital-output ratios), and whether or not 'growth is always perfectly balanced [i.e., full-employment growth]' (230–32). In fact, the participants <I>on both sides</I> of those debates were concerned with production systems in which substitution and capital-output variability occurred; and continuous full-employment growth was not entailed by recourse to orthodox, marginalist production functions, a point perfectly understood by the participants on both sides. -- Tony Aspromourgos </BLOCKQUOTE><P><B>Update (27 October 2014):</B> Added the Bernardo, Martinez, and Stockhammer reference. </P><P><B>Update (1 December 2014):</B> Added the Foster and Yates reference. </P>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com2tag:blogger.com,1999:blog-26706564.post-31280841729525455282014-10-17T08:16:00.000-04:002014-10-17T08:16:00.179-04:00r > g In A Steady State<B>1.0 Introduction</B><P>This post presents a model of distribution that Luigi Pasinetti developed. It is one of a family of models. Other important models in this family were developed by Richard Kahn, <A HREF="http://robertvienneau.blogspot.com/2008/10/kaldors-contributions-impressionistic.html">Nicholas Kaldor</A>, and Joan Robinson. These <A HREF="http://robertvienneau.blogspot.com/2012/01/nells-diagram-of-capitalist-economy.html">models</A> have been <A HREF="http://robertvienneau.blogspot.com/2007/01/post-keynesian-model-of-growth-and_07.html">extended</A> <A HREF="http://robertvienneau.blogspot.com/2007/01/post-keynesian-model-of-growth-and.html">in</A> <A HREF="http://robertvienneau.blogspot.com/2007/01/post-keynesian-model-of-growth-and_08.html">various</A> <A HREF="http://robertvienneau.blogspot.com/2007/01/post-keynesian-model-of-growth-and_10.html">ways</A> and presented in <A HREF="http://robertvienneau.blogspot.com/2006/09/non-teaching-of-empirical-superiority.html">textbooks</A>. One can see this family as having extended work by <A HREF="http://robertvienneau.blogspot.com/2010/08/why-income-equality-leads-to.html">Roy Harrod</A>, and as being related to the work of <A HREF="http://robertvienneau.blogspot.com/2007/03/michal-kalecki-one-of-greatest.html">Michal Kalecki</A> and even of <A HREF="http://robertvienneau.blogspot.com/2009/02/simple-and-expanded-reproduction.html">Karl Marx</A>. </P><B>2.0 The Model</B><P></P><B>2.1 Definitions</B><P>Consider a simple closed economy with no government. All income is paid out in the form of either wages or profits: </P><BLOCKQUOTE><I>Y</I> = <I>W</I> + <I>P</I>, </BLOCKQUOTE><P>where <I>W</I> is total wages, <I>P</I> is total profits, and <I>Y</I> is national income. Total savings is composed of savings by workers and by capitalists, where capitalists are a class whose members receive income only from profits: </P><BLOCKQUOTE><I>S</I> = <I>S</I><SUB>w</SUB> + <I>S</I><SUB>c</SUB></BLOCKQUOTE><P><I>S</I> is total savings. <I>S</I><SUB>w</SUB> is workers' savings, and <I>S</I><SUB>c</SUB> is capitalist savings. Profits are also split into two parts: </P><BLOCKQUOTE><I>P</I> = <I>P</I><SUB>w</SUB> + <I>P</I><SUB>c</SUB>, </BLOCKQUOTE><P>where <I>P</I><SUB>w</SUB> is returns on the capital owned by the workers, and <I>P</I><SUB>c</SUB> is the return on the capital owned by the capitalists. The behavior assumption is made that both workers and capitalists save a (different) constant proportion of their income: </P><BLOCKQUOTE><I>S</I><SUB>c</SUB> = <I>s</I><SUB>c</SUB> <I>P</I><SUB>c</SUB></BLOCKQUOTE><BLOCKQUOTE><I>S</I><SUB>w</SUB> = <I>s</I><SUB>w</SUB> (<I>W</I> + <I>P</I><SUB>w</SUB>) </BLOCKQUOTE><P><I>s</I><SUB>c</SUB> is the capitalists' (marginal and average) propensity to save. <I>s</I><SUB>w</SUB> is the workers' (marginal and average) propensity to save. The propensities to save are assumed to lie between zero and one and to be in the following order: </P><BLOCKQUOTE>0 ≤ <I>s</I><SUB>w</SUB> < <I>s</I><SUB>c</SUB> ≤ 1 </BLOCKQUOTE><P>Workers' savings are assumed to be insufficient to fund all the investment occurring along a steady-state growth path. </P><P>The value of the capital stock is divided up into that owned by the workers and by the capitalists: </P><BLOCKQUOTE><I>K</I> = <I>K</I><SUB>w</SUB> + <I>K</I><SUB>c</SUB>, </BLOCKQUOTE><P>where <I>K</I> is the value of the capital stock, <I>K</I><SUB>w</SUB> is the value of the capital stock owned by the workers, and <I>K</I><SUB>c</SUB> is the value of the capital stock owned by the capitalists </P><B>2.2 Steady State Equilibrium Conditions</B><P>Along a steady-state growth path, in this model, all capital earns the same rate of profits, <I>r</I>: </P><BLOCKQUOTE><I>r</I> = <I>P</I>/<I>K</I> = <I>P</I><SUB>c</SUB>/<I>K</I><SUB>c</SUB> = <I>P</I><SUB>w</SUB>/<I>K</I><SUB>w</SUB></BLOCKQUOTE><P>It follows from the above set of equations that the ratio of the profits received from the workers to the profits received by the capitalists is equal to the ratio of the value of capital that each class owns: </P><BLOCKQUOTE><I>P</I><SUB>w</SUB>/<I>P</I><SUB>c</SUB> = <I>K</I><SUB>w</SUB>/<I>K</I><SUB>c</SUB></BLOCKQUOTE><P>Likewise, one can find the ratio of total profits to the profits obtained by the capitalists: </P><BLOCKQUOTE><I>P</I>/<I>P</I><SUB>c</SUB> = <I>K</I>/<I>K</I><SUB>c</SUB></BLOCKQUOTE><P>The analysis is restricted to steady-state growth paths where the value of the capitalists' capital and the value of the workers' capital is growing at the same rate: </P><BLOCKQUOTE><I>S</I>/<I>K</I> = <I>S</I><SUB>c</SUB>/<I>K</I><SUB>c</SUB> = <I>S</I><SUB>w</SUB>/<I>K</I><SUB>w</SUB></BLOCKQUOTE><P>The ratio of profits to savings is the same for the economy as a whole and for workers: </P><BLOCKQUOTE><P>P</I>/<I>S</I> = (<I>P</I>/<I>K</I>)/(<I>S</I>/<I>K</I>) = (<I>P</I><SUB>c</SUB>/<I>K</I><SUB>c</SUB>)/(<I>S</I><SUB>c</SUB>/<I>K</I><SUB>c</SUB>) = <I>P</I><SUB>c</SUB>/<I>S</I><SUB>c</SUB></BLOCKQUOTE><P>Or, after a similar logical deduction for workers: </P><BLOCKQUOTE><P>P</I>/<I>S</I> = <I>P</I><SUB>c</SUB>/<I>S</I><SUB>c</SUB> = <I>P</I><SUB>w</SUB>/<I>S</I><SUB>w</SUB></BLOCKQUOTE><P>Along a steady-state growth path, planned investment, <I>I</I> equals savings: </P><BLOCKQUOTE><I>I</I> = <I>S</I></BLOCKQUOTE><B>2.3 Deduction of the Cambridge Equation</B><P>The following is a series of algebraic substitutions based on the above: </P><BLOCKQUOTE><I>P</I>/<I>I</I> = <I>P</I>/<I>S</I> = <I>P</I><SUB>c</SUB>/<I>S</I><SUB>c</SUB> = <I>P</I><SUB>c</SUB>/(<I>s</I><SUB>c</SUB> <I>P</I><SUB>c</SUB>) = 1/<I>s</I><SUB>c</SUB></BLOCKQUOTE><P>Or: </P><BLOCKQUOTE><I>P</I> = (1/<I>s</I><SUB>c</SUB>) <I>I</I></BLOCKQUOTE><P>The share of profits in national income is determined by the savings propensity of the capitalists and the ratio of investment to national income: </P><BLOCKQUOTE>(<I>P</I>/<I>Y</I>) = (1/<I>s</I><SUB>c</SUB>) (<I>I</I>/<I>Y</I>) </BLOCKQUOTE><P>Recall that the rate of profits is the ratio of profits to the value of capital: </P><BLOCKQUOTE><I>r</I> = <I>P</I>/<I>K</I> = (1/<I>s</I><SUB>c</SUB>) (<I>I</I>/<I>K</I>) </BLOCKQUOTE><P>Recognizing that <I>I</I>/<I>K</I> is the rate of growth, <I>g</I>, one obtains the famous <I>Cambridge equation</I>: </P><BLOCKQUOTE><I>r</I> = <I>g</I>/<I>s</I><SUB>c</SUB></BLOCKQUOTE><P>As long as the capitalists consume at least some of their income, the rate of profits is greater than the rate of growth along a steady-state growth path. And along such a path the share of income going to profits will be constant. </P><B>3.0 Discussion</B><P>If one assumes given investment decisions, the Cambridge Equation tells us what rate of profit is compatible with a steady state growth path in which the expectations underlying those investment decisions are satisfied. </P><P>Consider two steady states in which the same rate of growth is being obtained. Suppose that along one path workers have a higher propensity to save. Within broad limits, this greater willingness to save among workers has no effect on determining either the share of profits in income or the rate of profits. Only the capitalists' saving propensity matters for the steady state rate of profits, given the rate of growth. Would a capitalist economy have a tendency to approach such a growth path, given a sufficient length of time? I think such stability would entail the evolution of institutions, conventions, the labor force, and what is seen as common sense, including among dominant political parties. </P><P>The above model might have some relevance to current political economy discussions <A HREF="http://crookedtimber.org/2014/10/13/r-g/">elsewhere</A>. </P> Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com0tag:blogger.com,1999:blog-26706564.post-67159070719034792482014-10-14T15:26:00.000-04:002014-11-17T07:30:02.876-05:00Jean Tirole, A Practitioner Of New Industrial Organization<P>I have occasionally <A HREF="http://robertvienneau.blogspot.com/2014/01/impact-of-piero-sraffa-on-industrial.html">summarized</A> <A HREF="http://robertvienneau.blogspot.com/2013/01/selected-principles-of-microeconomics.html">certain</A> <A HREF="http://robertvienneau.blogspot.com/2010/06/formalism-in-economics.html">aspects</A> of microeconomics, concentrating on markets that are not perfectly competitive. Further developments along these lines can be found in the theory of Industrial Organization. </P>One can distinguish in the literature two approaches to IO know as <I>old IO</I> and <I>new IO</I>. Old IO extends back to the late 1950s. Joe Bain and Paolo Sylos Labini laid the foundations to this approach, and they were heralded by Franco Modigliani. I have not read any of Bain and only a bit of Sylos Labini. Sylos was a Sraffian and quite critical of neoclassical economics. He also had interesting things to say about economic development. <P>As I understand it, new IO consists of applying game theory to imperfectly competitive and oligopolistic markets. I gather new IO took off in the 1980s. Jean Tirole, the winner of this year's "Nobel" prize in economics, is a prominent exponent of new IO. </P><P>One can tell interesting stories about corporations with both old IO and new IO. For example, Tirole has had something to say about vertical integration which, based on what I've read in the popular press, might be of interest to me. (Typically, when I explore the theory of vertical integration, following Luigi Pasinetti, the integration is only notional, not at the more concrete level of concern in IO.) </P><P>I wonder, though, whether economists can point to empirical demonstrations of the superiority of new IO over old IO. Or have economists studying IO come to embrace new IO more because of the supposed theoretical rigor of game theory? Are specialists in IO willing to embrace the indeterminism that arises in game theory, what with the variety of solution concepts and the existence of multiple equilibria in many games? Or do they insist on closed models with unique equilibria? </P><B>References</B><UL><LI>Franco Modigliani (1958). New developments on the Oligopoly Front, <I>Journal of Political Economy</I>, V. 66, No. 3: pp. 215-232.</LI></UL><P><B>Update (same day):</B> Corrected a glitch in the title. Does this Paul Krugman <A HREF="http://krugman.blogs.nytimes.com/2014/10/14/jean-tirole-and-the-triumph-of-calculated-silliness/">post</A> read as a direct response to my post? </P>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com1tag:blogger.com,1999:blog-26706564.post-34718944115530232202014-09-30T15:25:00.000-04:002014-10-09T08:06:14.952-04:00Noncommunicating Literatures?<P>During the 20th century, a number of economists more or less independently developed ideas associated with input-output analysis, activity analysis, modeling the economy as a self-sustaining circular flow, and the revival of classical political economy. I think of: </P><UL><LI>Leonid Kantorovich: The Soviet economist who shared the 1975 Nobel Memorial Prize in Economic Sciences with Tjalling Koopmans.</LI><LI>Wassily Leontief: Always emphasized developing an empirically operational version of these ideas.</LI><LI>Father Maurice Potron: I stumbled across two references to him. I know nothing otherwise about his <A HREF="http://www.amazon.com/The-Analysis-Linear-Economic-Systems/dp/041574654X">work</A>.</LI><LI>Walter Isard: Extended input-output analysis to regional economics.</LI><LI>Richard Stone: Developed the idea of a Social Accounting Matrix and conventions for national income accounting.</LI><LI>Jacob Schwartz: Criticized the mainstream economics of his time on the basis of linear economic models.</LI><LI>Piero Sraffa: Criticized the mainstream economics of his time on the basis of linear economic models.</LI><LI>John Von Neumann: A mathematician, not an economist.</LI></UL><P>I wonder how many make connections between the scholarly literature building on the work of each of these researchers. I am not at all sure anybody explicitly and consciously built on Potron or Schwartz. </P><B>References</B><UL><LI>Wassily W. Leontief (1936). Quantitative Input and Output Relations in the Economic Systems of the United States, <I>Review of Economic Statistics</I>, V. 18, N. 3 (Aug). pp. 105-125.</LI><LI>Walter Isard (1951) Interregional and Regional Input-Output Analysis: A Model of a Space-Economy, <I>Review of Economics and Statistics</I>, V. 33, No. 4 (Nov.): pp. 318-328.</LI><LI>Jacob T. Schwartz (1961). <I>Lectures on the Mathematical Method in Analytical Economics</I>, Gordon and Breach.</LI><LI>Piero Sraffa (1960). <I>Production of Commodities by Means of Commodities: A Prelude to a Critique of Economic Theory</I>, Cambridge University Press.</LI><LI>J. Ricard N. Stone (1966). The Social Accounts from a Consumer Point of View, <I>Review of Income and Wealth</I>, V. 12, Iss. 1 (Mar.): pp. 1-33. [I HAVEV'T READ THIS OR ANYTHING ELSE BY STONE]</LI><LI>John von Neumann (1945-1946) A Model of General Economic Equilibrium, <I>Review of Economic Studies</I>, V. 13, No. 1: pp. 1-9.</LI></UL>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com5tag:blogger.com,1999:blog-26706564.post-66356699857774091652014-09-19T08:10:00.000-04:002014-10-01T06:46:53.428-04:00Hayek Not Opposed To Keynes On Political Principle<P>With characteristic cheerful carelessness, Noah Smith <A HREF="http://www.bloombergview.com/articles/2014-09-11/how-keynes-became-a-dirty-word">misinforms</A> hapless <I>Bloomberg</I> readers: </P><BLOCKQUOTE>"Friedrich Hayek tried to argue against Keynes' theories, but for whatever reason, he lost the debate among economists in the 1930s. But Hayek would have the last laugh, because in his book, 'The Road to Serfdom,' he attacked Keynes from a very different angle. Instead of saying Keynes' theories were wrong, Hayek prophesied that Keynesian stabilization policies would lead down the slippery slope to totalitarianism." </BLOCKQUOTE><P>As a matter of fact, Hayek said nearly the opposite: </P><BLOCKQUOTE>"There is, finally, the supremely important problem of combating general fluctuations of economic activity and the recurrent waves of large-scale unemployment which accompany them. This is, of course, one of the gravest and most pressing problems of our time. But, though its solution will require much planning in the good sense, it does not - or at least need not - require that special kind of planning which according to its advocates is to replace the market. Many economists hope, indeed, that the ultimate remedy may be found in the field of monetary policy, which would involve nothing incompatible even with nineteenth-century liberalism. Others, it is true, believe that real success can be expected only from the skilful timing of public works undertaken on a very large scale. This might lead to much more serious restrictions of the competitive sphere, and, in experimenting in this direction, we shall have to carefully watch our step if we are to avoid making all economic activity progressively more dependent on the direction and volume of government expenditure. But this is neither the only nor, in my opinion, the most promising way of meeting the gravest threat to economic security. In any case, the very necessary efforts to secure protection against these fluctuations do not lead to the kind of planning which constitutes such a threat to our freedom." -- Frierich A. Hayek, <I>The Road to Serfdom</I> (1944), Chapter IX. </BLOCKQUOTE><P>Both Hayek and Keynes drew on nineteenth-century Liberalism. They agreed that the inherited lines limiting government action needed to be redrawn. Keynes said as much in the 1920s, in his essays republished in <I>Essays in Persuasion</I>. Hayek's reference above, to the "timing of public works" is to Keynes' ideas. Keynes doubtless would have redrawn the lines more broadly then Hayek. But Hayek explicitly says above that Keynes' approach is neither necessarily a threat to freedom, nor a station on the way to totalitarianism. Hayek says his differences with Keynes are pragmatic, a dispute over what is likely to be effective. </P>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com1