Home > The Economics Profession > “Inapplicable operations on ordinal, cardinal, and expected utility”

“Inapplicable operations on ordinal, cardinal, and expected utility”

from Issue no. 63 of the real-world economics review

Inapplicable operations on ordinal, cardinal, and expected utility
Jonathan Barzilai   [Dalhousie University, Canada]      download pdf

Editor’s note:

This short paper was originally submitted to World Economics Review, where under its online open review it was for a year subjected to voluminous high calibre critique and author response (available here). As the first reviewer noted: “If the author is right, a substantial part of orthodox economics has to be rejected on purely formal grounds”. The paper’s arguments turn on the application of abstract algebra, a branch of mathematics in which we economists are rarely fluent. The paper asserts:

  1. Hick’s and Samuelson’s applications (and those based thereon) of differentiation to ordinal utility are founded on mathematical errors.
  2. Expected utility’s scale construction rule is self-contradictory.

By publishing Jonathan Barzilai’s paper in the RWER, it is hoped that one or more mathematicians will bring their expertise to bear on its argument and that the high calibre consideration of the paper by economists will continue in public view. To this end, this post has been placed so that people may comment on the paper. Only comments of an academic nature and directed primarily to the paper will be posted. 

  1. Claude Hillinger
    March 27, 2013 at 2:23 pm

    In econometric work on consumer demand, as well as in other areas, economists do not use ordinal functions (whatever that may be). They use quite ordinary cadinal differentialble functions, such as Cobb- Douglas or CES. The commitment to ordinal utility merely means that no meaning is attached to the numerical value of utility implied by such a function. The fundamental statement in this area is that any positive monotone transformation of a given utility function implies no change in predicted observeable behavior.

    • merijnknibbe
      March 27, 2013 at 3:45 pm

      Did anybody ever observe a positive monotone transformation of a given utility function? Anybody? Ever?

      • Claude Hillinger
        March 27, 2013 at 4:12 pm

        y(x)=f(g(x), f’>0. Voilà

      • merijnknibbe
        March 28, 2013 at 3:23 pm

        I mean: *observe*. That is: watch it in real life.

  2. March 30, 2013 at 2:25 pm

    “By publishing Jonathan Barzilai’s paper in the RWER, it is hoped that one or more mathematicians will bring their expertise to bear on its argument and that the high calibre consideration of the paper by economists will continue in public view. … Only comments of an academic nature and directed primarily to the paper will be posted.”

    So, referring to a background in experimental physics/electronic communication, computing and control systems/reliability, quality, project and budgetary control/the development of non-numerical programming methods (using indexed data and logic) and automation of mathematical procedures, I’m a right-brained practical scientist with life-long interests in the history and development of philosophy and technology of science, mathematics and physiological psychology, not a left-brained academic mathematician. By this I mean my aims are practical rather than academic and (as was required in 1818 of engineers) “like an interpreter between two foreigners [here the academic mathematician and the technician, I], must understand the language of both”. Does that mean my comments will not be published?

    “If the author is right, a substantial part of orthodox economics has to be rejected on purely formal grounds”. [Egmont Kakarot-Handtke. March 19, 2012].

    My reaction to Barzalai’s paper was that basically he IS right, but 45 pages of wordy comment, largely comprising claims and counter-claims, left me aware that his definitions amount to claims which will NOT lead to K-H’s “rejection on purely formal grounds” because they are not “purely formal”: they are expressed in language which is partly conventional and – even abstracting all content – determines points of view. Even if Utopia had existed, today’s “adult” skeptics won’t believe in what they haven’t seen for themselves.

    On the other hand, being familiar with parts of mathematics Barzalai seems not to be, I can use four-level Algol68-R to define the meanings of operations like ‘addition’ for different interpretations of numerals: notably clock numbers, decomposable by Pythagoras’s Theorem and Fourier’s Transformation, for which differentiable units rotate at different speeds and differentiation is defined – as in Newton’s tangential method – by rotation through a right angle (i.e., in complex number mathematics, by i). Given the Big Bang represents the 0 of time and the dynamic Fourier 1 a complete clock hand rotation, one has the language to explain a compass having four cardinal points, language four levels of meaning, logic four types (input/output as well as indexical left/data mapping right), four distinct types of linguistic, economic etc. functions humans and computers can learn, perform, fail at and become specialists in, and a sequence of points (representing the evolution of species) at which a cycle of time repeated and the same symbol has two meanings, as when 12 o’clock represents both the end of one hour and the beginning of the next.

    I see the transition from the classical to the neo-classical marked recognition of the completion of the evolution of economics to reliance on trust (“I promise” rather than barter involving physical securities), and the beginning of the evolution of a dishonest chremistatics – i.e. money making based on usury, gambling, insurance and (recently) derivatives – while still calling itself economics but (as in that Citybank leak, RWER blog, June 20, 2011) trivialising as a thing of the past the self-regenerating household economy of children, men, women and elders based on cooperation and recompense for/rewarding of service. We can now see that our trust has been misplaced: indeed, it has been destroyed by a left-brained chremistatic ethos of one-sided competitiveness.

    So, as a practical man I appreciate Barzalai’s argument but doubt its effectiveness. To be constructive, I draw attention to directed motion having preceded his Euclidian space, yet vectors as he defines them (as against how engineers define them) are unable to account for direction and hence circularity, closure, localisation and limitation.

    As we were reminded in an accompanying paper, Keynes famously announced his revolution with a reference to Euclid: “Yet, in truth, there is no remedy except to throw over the axiom of parallels and to work out a non-Euclidean geometry. Something similar is required today in economics”. (Keynes, 1973, p. 16) “This in turn would have required some sort of non-Euclidean axioms, that is, a bit more formalization than Keynes was prepared to do himself.” (Kakarot-Handtke, 2012).

    Precisely. Our human home is not an infinite Euclidian plane measurable against pairs of Cartesian coordinates but, like the universe – Hubble’s expanding bubble – the 3-dimensional skin of a sphere measured in degrees along circular coordinates, i.e. the equator and another through Greenwich and the poles. Both latitude and longitude are ordinal but only longitude is also a one-dimensional Barzalai vector. For dimensionality to be preserved latitude has to remain ordinal: a non-linear two-dimensional angle not metrically independent of longitude. Taking the Big Bang as one axiom and the speed of light as another (not as assumptions but as conclusions reached as science became able to examine evidence from further and further back in time), these coordinates and some fortuitous discoveries about circuits have enables me to perceive and explain remarkably simply the four phases of each successive stages of evolution and key interactions such as gravity, polarised magnetism, bisexual reproduction, split-brain consciousness and the blindness of the vast majority of successful academics and political pedagogues, who use more or less only the verbal (left) side of their brain: like mathematicians, efficiently manipulating indexes without reflecting on the data. It seems Professor Roger W Sperry received the Nobel Prize in 1981 for his innovative studies of the differing blindnesses of surgically split brains (Edwards, 1992), but G K Chesterton (1904, 1908) had already reached the same conclusions via earlier scientific observation of stroke victims, and St Paul had – in principle – done so long before him (1 Cor 12, a.d. 57).

    It is said “a diagram is worth a thousand words”, but with the iconic lines and angles of Euclidian geometry hidden in the arbitrary symbols and algebraic form of point set topology and Cartesian co-ordinate geometry, minimally complex (i.e. four component, four loop) circuit diagrams have become as indispensible for honest economics as they have been (despite academics) for electronic theorising and pedagogy. The word ‘circle’ is meaningless until one has seen one, used a compass to draw one or recognised one as a special case of an ellipse or topological circuit (Sawyer, 1955). I can show you on the back of an envelope how economies work and Barzalai’s “all male” family tree logic doesn’t, but endless algebra and words-only comment are likely to be counter-productive when what is needed is a topological circuit diagram (Taylor, 2005) and willingness to learn how economics can be interpreted with the aid of it.

    References.

    ALGOL68-R: P M Woodward and S G Bond, “Algol 68-R Users Guide”, 2nd edn 1974, HMSO.
    [http://www.fh-jena.de/~kleine/history/languages/Algol68R-UserGuide.pdf]. The four levels of Fregeian “sense and reference” are values, variables, modes [of interpretation] and procedures. Interpreted as logic circuit switch settings, the latter enable computers to actively process and regenerate the other levels.

    CHESTERTON, 1904: G.K.Chesterton: “G F Watts” [an art-trained critic’s appreciation], Duckworth, 1904, pp. 39-49. [http://archive.org/details/gfwattschesterto00chesiala].

    CHESTERTON, 1908: G.K.Chesterton: “Orthodoxy”, [i.e. “Heterodoxy” relative to our left-brained culture], “without any doubt the greatest work of English literature this century” [Milward, GKC Review, Nov 1981]; 1908, Ch.2, “The Maniac”. [http://www.gutenberg.org/files/16769/16769-h/16769-h.htm].

    EDWARDS, 1992: Betty Edwards: “Drawing on the Right Side of the Brain: a Course in Enhancing Creativity and Artistic Confidence”, BCA, British edition 1992, pp. xi, 26-43.

    KAKAROTE-HANDTKE, 2012: Egmont Kakarot-Handtke, “Crisis and methodology: some heterodox misunderstandings”, real-world economics review, issue no. 63, 25 March 2013, pp. 98-117. [http://www.paecon.net/PAEReview/issue63/kakarot-handtke63.pdf]. A truly delightful paper, but at p.99, wrong to follow Popper on methodology. My comments above refute his claim that “heterodoxy has no promising methodological alternative to offer”. Heterodoxy is heteronomous, and my type is offering Critical Realist philosophy, SSADM structured systems analysis and Algol-68R scientific modelling.

    CITIGROUP, 2011: RWER blog. [https://rwer.wordpress.com/2011/06/20/from-november-2010-citigroup-attempts-to-disappear-its-plutonomy-report-2/]

    SAWYER, 1955: W W Sawyer: “A Prelude to Mathematics [: On Growing Mathematicians]”, which c.1973 my colleague and I independently concluded “was the best book on mathematics we had ever come across”; Pelican, 1955, p.28. [http://archive.org/details/PreludeToMathematics].

    TAYLOR, 2005: D J Taylor: “From Theorising to Modelling: Mondragon Cooperatives as economic ‘models’ “, in IACR 9th Annual Conference Papers, University of Western Sydney, 10-12 July 2005. Published internally, so from the abstract: “It is argued that the basic theory of economics is analogous to electrical theory, which is based on a readily visualisable topological circuit diagram form of mathematics. The two uses of electrical circuits – to distribute power and to convey information “piggy-backing” on power – suggest two different approaches to economic theorising”. The information system approach was modelled using SSADM, the Structured Systems Analysis and Design Method.

    • March 30, 2013 at 3:09 pm

      THANK YOU, davetaylor1.
      And thanks to RWER for allowing your comment to be recorded.
      “***** “Believe nothing merely because you have been told it…But whatsoever, after due examination and analysis,you find to be kind, conducive to the good, the benefit,the welfare of all beings – that doctrine believe and cling to,and take it as your guide.”- Buddha[Gautama Siddharta] (563 – 483 BC), Hindu Prince, founder of Buddhism.
      Who among all those listed on this page as ‘contributors’ would challenge,improve or endorse….”.I see the transition from the classical to the neo-classical marked recognition of the completion of the evolution of economics to reliance on trust (“I promise” rather than barter involving physical securities), and the beginning of the evolution of a dishonest chremistatics – i.e. money making based on usury, gambling, insurance and (recently) derivatives – while still calling itself economics but (as in that Citybank leak, RWER blog, June 20, 2011) trivialising as a thing of the past the self-regenerating household economy of children, men, women and elders based on cooperation and recompense for/rewarding of service. We can now see that our trust has been misplaced: indeed, it has been destroyed by a left-brained chremistatic ethos of one-sided competitiveness.”

      • davetaylor1
        March 31, 2013 at 7:56 am

        Thank YOU, Lucky.

        Dear learned CONTRIBUTORS to RWER and its blog, may I draw attention to the question Lucky has buried in his comment:

        “Who among all those listed on this page as ‘contributors’ would challenge, improve or endorse” my characterisation of the difference between classical and neo-classical economics?

        I should have added the thought that the one response to the perverted evolution of economics based on trust is to correct its errors in a more or less Keynesian fashion. The other (as Kakarote-Handtke has argued and I have agreed) is to “scrap the lot and start again”. He still believes in the relevance of equations and wanted to correct them; my belief is that we need to start at a deeper level where information science has evolved from physical communication technology via a Shannon observation as fortuitous as the falling of Newton’s apple.

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