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3 Mathematical Economics

3 Rybicki W.

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ON THE FORMALIZATION AND INTERPRETATION OF THE NOTION OF 'EFFICIENCY' - FURTHER EXAMPLES AND COMMENTS (O formalizacji i interpretacji pojecia efektywnosci - dalsze przyklady i komentarze)

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Rybicki W.

nr 6(13)

103-126

The article is devoted to revealing further kinds of the meaning of the term 'efficiency' (and related notions), consequently setting them in mathematical frames. First of all the so called envelope-type efficiency is introduced. This notion is illustrated by several examples derived from the elementary topology, Bayesian statistics, mathematical economics and primer of financial engineering. It seems that the above examples do reflect the essence of this idea in the best possible picture. The next proposition concerns the type of efficiency which was called 'collective-type efficiency'. It turns out that the reasonable compromise is 'better than the best solution' (even in the Nash sense). The quite good class of examples are provided by problems derived from the famous 'prisoner's dilemma' and exploitation of common resources. At the end of the paper some complementing thoughts are indicated - in a loose form. They concern the principal conflict between efficiency and equity as well as the problem of economic behaviour in the sphere of scientific research (the balance between two factors: 'erudite components' and 'creative potentials' of the researchers).

Arbitrage in economics and elsewhere – facts well known and less known (three papers on arbitrage ideas, modeling and pricing – yesterday, today and tomorrow) – introduction to the series

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Rybicki W.

2011

nr 7 (14)

169-172

The presented series of articles on arbitrage theories and their methodological aspects consist of three papers entitled as follows: I. The Primer on Arbitrage Conceptions in Economics: Their Logics, Roots and Some Formal Models (Historical and Bibliographical Notes). II. Mathematics of Financial Arbitrage: From Algebraic Geometry at the Turn of the 19th and 20th Centuries to Modern Martingale (Generalized) Considerations III. The Arbitrage in Stochastic Finance, Social Choice Theory and Macroeconomics. The articles are devoted to present – in a historical perspective – the basic ideas and “metamorphoses” of the notion and role of an arbitrage (originated as a kind of clever and rational speculation – the last word is used in a “neutral”, not pejorative sense) as well as to point out its various, important connotations (not merely in finance or even economics) and to demonstrate some mathematical inevitable technicalities, reflecting, in fact, the logical essence and the modern view of the arbitrage (and non arbitrage) conditions.

The primer on arbitrage conceptions in economics: their logics, roots and some formal models (historical and bibliographical notes)

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Rybicki W.

2011

nr 7 (14)

173-197

The paper makes up the first part of a larger study devoted to arbitrage ideas, models and pricing methodology in spirit of “no arbitrage” (or fairness or transparency) demands.The work – as a whole – is entitled “Arbitrage in Economics and Elsewhere – Facts Well Known and Less Known” and consists of three papers. In the present essay we intentionally interweave “loose (informal) variations on themes” (of arbitrage theories, their applications and connotations) with (brief) demonstrations of selected formal models and some more rigorous mathematical technicalities. Some efforts are made to highlight significant economic aspects as well as to reveal a piece of mathematical “machinery” hidden behind the stories told. Nevertheless, the introductory character of the current paper causes the descriptive, philosophical and historical elements to prevail: we invoke very old roots such as Aristotle‟s or Aquinata‟s thoughts and then follow Cournot, Walras and Keynes works, up to the crucial paper of Miller, Modigliani. Along the way the very deep considerations on the coherency of subjective probability systems are mentioned – “the probabilistic core” of an arbitrage/no arbitrage questions (thoughts of Ramsey and de Finetti). Subsequently, the basics (finite state-space) of the modern, martingale (no arbitrage) modeling (originated by Harrison, Kreps, Pliska) is presented, as well as the “factor-type” schema of the arbitrage pricing theorem (Ross‟s conception). The role played by the supplemented bibliography should be also pointed out. It significantly enters the planned communication. The author‟s aim was to provide the (selected) basis, and “vocabulary” which will be useful for reading the entirety of the “trilogy” – the presented foreword really constitutes a kind of “a bibliographical note”.