Utility and Measurement (part 3a)

NOTICE

• May contain errors: DO NOT TRUST THE FOLLOWING
• For this vast and vague subject, even a course survey is not presented
• Sorry for Jeremy Bentham and ethical thinkers all time who take heterogeneity seriously, little treatment given for social contexts

Far Historical Landscape of Decision and Social Choice Theory

• There is impressive contrast between slow development of decision (voting) theory, which treats ordinal utility, or preference, and fast development of utility theory, which treats mainly cardinal utility
• Despite that we compare one with another everyday, and discrete and abstract argument of preference looks well intuitive, ordinal utility could not be well understood without complete foundation of measure theory, topology and set theory. Any interesting (or dismal) choice in reality faces on opposition and uncertainty, both requires utility calculus based on sets and measures, and defined utility make various such conditions comparable
• Bernoulli(1738, trans. 1954) considered St. Petersburg paradox and gave modern notion of risks and utility with uncertainty, left unconsidered before von Neumann-Morgenstern(2/e, 1947)
• (aside) Essentially also by von Neumann, together with some other mathematical abstractions, general topology were introduced to economics
• Condorcet(1785), known for voting paradox, was the first systematic treatise on decision theory, using order-theoretic argument built on probability theory. Simply it was too early for the human race in pre-Bourbaki age
• Essentially no disciplinary landmark before Arrow(1951, 1963)
• For detailed exposition, see Duncan Black(1958, 1998)
• Game theory give a way to describe operations to partial-ordering of consistent social preference. Discrete conception of preference and game made microeconomic arguments applicable to areas where no explicit or continuous quantity and price is given: these span for fundamentals of several disciplines in economic (off-market competition), social and political studies.

TODO

• write the latter half of part 3 (may treat development of utility theory and unnamed open sets inside indifference curves)

Price Index, Monetary Redenomination and Discounting (half-baked)

NOTICE

• still half-baked: DO NOT TRUST THE FOLLOWING

Outlines

• Sometimes total reform of numerical structure of economic quantities needed
• Components of consumer price indexes and of market indexes like Dow Jones need to be changed time after time: there was no GOOG before 2004, no AAPL before 1980. Such components are basis in certain vector space but can not stay adequate over time
• Even monetary redenomination is needed rarely, usually as a process in aftermath of hyperinflation
• In econometrics, time-series data require unit root test. Time-evolution of such data could be non-unitary and should be detrended before calculating correlation. Two pro-cyclical data would have unit root in common, which is equal to inflation rate
• It might not so inadequate to call such act of reform of economic quantities system as renormalization
• Discounting factors, or inflation rates might be called as curvature
• There's no constant approximate wealth functional (and no orthonormal basis) even after discounting of inflation rate (was the key issue in Cambridge capital controversy)
• In other words, space of goods grows: infinite dimensional, not compact and non-separable
• Hence, maximization of certain wealth functional, or intertemporal substitution, might not make sense in long-run

TODO

• find physical analogues of Paasche, Laspeyres, and etc.
• explain what real interest rate is (or is not), and when Irving Fisher's approximation i = r + p (and neutrality of money) breaks
• examine index number requirements with the theory of measurement
• explore relations to FPTs and other well-known results in functional analysis (may take years)

On Personal Matters and Forthcoming Posts

Since I'm living near Tokyo and for ongoing Fukushima reactor crisis as of 19th March 2013 and curious arrhythmia, high blood pressure and thyroid enlargement noticed after June 2012, I have little hope to survive coming five years or even less. Around home and office, almost everyday I hear ambulance siren. My wife daily evacuates slightly reddish mucus and I don't know what to impute, her vice of smoking or Radioaktivität in the air for you and me.

In forthcoming posts, treatment of these subjects is planned.

• As previously annonced, Utility and Measurement part 3, treating history of utility theory in economics and interrelation with topology, order theory and measure theory
• von Neumann's contribution to social sciences and to other diciplines
• Hamiltonian dynamics and Samuelson's economics

Sorry for irregular grumbling.

Utility and Measurement (part 2 beta)

NOTICE

• May contain errors: DO NOT TRUST THE FOLLOWING
• just a survey, no new claims intended
• Blame my wife if this post is more insufficient than previous posts :)

History in Physics

• Like other metaphysical categories (space, time, whole vs. part), distinctions between quality (or intensity) and quantity (or extension) is soon relativized after clarification
• I don't know the very first appearance of "Intensive" and "Extensive" in thermodynamics literature
• Gibbs(1876) introduced "Intensive" concept, but no inclusive concept for extensive values: he just enumerated values which are proportional to volume in equilibrium
• Helmholtz(1887) explicitly referred to old metaphysicians' conception of "Extensive" and "Intensive" contrast, respectively adding "Mr. P. du Bois-Raymond named former as linear magnitude and latter as non-linear"
• Hölder(1901) presented axioms for theory of measurement
• Duhem(1911) assured mediæval roots of this metaphysical distinction (mentioned in part 1)
• Textbooks of thermodynamics: In Duhem(1902) and Planck(1903) these concepts do not appear as such
• (Added Sep 17 2013) Richard C. Tolman proposed terminology of "Extensive" and "Intensive" properties in 1917, as reported in O. Redlich(1970)
• Enswiler(1921), as an product of transitional period, use "intensity factor" and "distribution factor"
• Enswiler(1921), as an product of transitional period, use "intensity factor" and "distribution factor"
• Fermi(1937, 1956) avoids use of "Intensive-Extensive" and use homogeneity (of 0th or 1st degree) instead

History in Mathematics

• After Helmholtz and Hölder, it was widely understood that extensivity is additivity
• Measure in general needs to be built on set theory. Affected by Cantor, Henri Lebesgue and his fellow French researchers tried to define measure
• After WW2, from a order-theoretic viewpoint, Dana Scott and Patrick Suppes contributed to theory of measurement (incomplete differentiation, non-monotonic change of variables, or projection, causes information loss - probably that's the motivation for domain theory)

TODO

• detailed bibliography
• or total refinement of Math part
• Scottish Metaphysician Sir W.Hamilton (1788–1856), who appear in OED entries "Intensive" and "Extensive" is not Irish Physicist Sir William Rowan Hamilton (1805–1865)
• preview for forthcoming: How economists once abandoned ordinal/cardinal utilities and still use them today? Stay tuned

Utility and Measurement (part 1)

NOTICE

• May contain errors: DO NOT TRUST THE FOLLOWING
• just a survey, no new claims intended

Least Philosophy

• Affection: In Kantian philosophy, Ding an Sich (thing as itself) is not accesible to human mind. Things just affect.

History in Philosophy

• (Pierre Duhem's finding) mediæval French philosopher Nicole Oresme's conception: latitudo (width), or intensio, and longitudo (length), or extensio
• In Cartesian philosophy, followed by Leibnitz and Spinoza, extension is very central concept (if interested to the linkage between Descartes and mediæval thoughts, Étienne Gilson's Index might be helpful)
• Pascal's Pensées: mysterious criticism against Cartesian way (algebraic models): "useless and uncertain"
• Cartesian coodinate is one of Leibnitz' inventions (putting aside similar ideas independently used). Another one is optimism
• Kant's Critique of Pure Reason: Euclidian space (space of extensive values) as a priori, perception and intension (to be written)
• Hegel's Orbits of Planet: over 200 years after Principia, argued against Newtonian synthesis of theories of conservative (Keplerian) system and dissipative (Galilean) system
• Common underlying insight of Pascal's and Hegel's (and later Duhem's) critiques (even if they were not aware): How the nature decide directions of changes?
• Leibnitz' solution was optimism, but it was insufficient then
• Vitalism was also insufficient
• Energetics was needed, but there was no thermodynamics yet
• pre-thermodynamical investigations in Hegel's Science of Logic: (1) Instant freezing of supercooled water as an example of transition from quantity to quality, (2) confused and "contradict" as usual, noticed on the fact that intensive values can always be represented as extensive values (example: temperature or pressure and length of mercury)

TODO

• clarify Kant's contributions with footnote about Deleuze on intensity and affection
• clarify relation to Bergson's "sugar melt" argument in Creative Evolution

On Possible Formalism of Austrian Economics

NOTICE

• Never take seriously and DO NOT TRUST THE FOLLOWING
• "Austrian Economics" in this post might share nothing with historical Austrian Economics

Least of Hilbert Space

• space of superposed harmonics with different phases

History (too partial and biased)

• Menger(father): originator, one of the marginal revolutionists
• Böhm-Bawerk: "wavy" view of economic reality
• Wieser: imputation (= decomposition of wealth)
• Mises(Lutwig): (Sorry, I know nothing about him)
• Hayek: "pure" capital theory and business cycle
• Schumpeter: rotation of capital and innovators' roles
• Menger(son): Brouwer's student, sought axiomatics for economics
• Morgenstern: collaborator of von Neumann and coauthor of TGEB

Possible Formalism

• enterprises and national economies as fields
• business cycles and prices as wave functions
• portfolio as state vector
• term structures as spectra
• capital and social wealth as functional, which could be decomposable
• quantities of bonds, or elements of the state vector, should be chosen for maximizing certain wealth functional (fixed point theorems assure that)
• risk and danger as hypothetical extremely long-term (infrared) bond with somewhat uncertain remaining duration (wave length) and quantity, which insurance companies might need to cover their contracts
• inflation rate, or "mean" return as curvature on timeline

TODO

• define equilibrium interest rates and explain economic meaning of eigenvalue
• complete explaination of economic meaning of FPTs in relation to Austrian understanding of "equilibria"
• treat consumption and surplus, not as depreciation of laborers' capital
• show similarity and difference to historical Austrian Economics
• show similarity and difference to pre-crisis mainstream macroeconomics
• show similarity and difference to modern finance

Matrices in Modern Economics: a sketch

NOTICE

• May contain errors: DO NOT TRUST THE FOLLOWING
• Strange abbreviations are intentional

Least of Linear Algebra

• Sylvester's criterion: real symmetric matrix is positive definite iff all leading principal minors > 0 (corresponds to sequence of variational approximation of multivariable function through its Jacobian matrix)

History (not in chronological order)

• one-sector growth model (no use of matrices): Malthus' EPP
• two-sector growth models (implicit use of linear algebra): Quesnay's TE and Marxian reproduction scheme in DK
• n-sector extensions of reproduction scheme: Sraffa's PCMC and Leontief's IOE
• total differentiability condition of cost and utility functions (Sylvester's criterion): Mathematical Appendix of Hicks' VC
• skew-symmetric (alternative) matrix: payoff matrices of two-player zero-sum games in von Neumann-Morgenstern's TGEB

TODO

• econometric models after Frisch
• modern dynamic models after Samuelson
• explain why no complex values allowed (or, why only "Hermitian operators" operating on "coherent states" allowed in dynamic models)
• other uses?