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1

SOME REASONS FOR WHICH WE TELL STUDENTS OF ECONOMICS ABOUT MATRICES (Kilka powodow, dla ktorych opowiadamy studentom ekonomii o macierzach)

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Rybicki W.
Didactics of Mathematics
|
2010
|
issue 7(10)
109-126
EN
In the paper we consider a role which a matrix plays in the educational process of students of economics (as a notion, a symbol of a mathematical operation as well as a numerical tool). We remind that matrices and determinants appear systematically in courses of mathematics and related subjects. They help to model and solve various significant problems of econometrics (wide sense) and operation researches. It is worth noting, howe-ver, that we make use of matrix notation in our lectures on microeconomics and macroeconomics. The paper initiates the series of three 'didactical' articles devoted to matrices. So it also plays a role of some kind of introduction to the subject. The article may be divided, in a natural way, into two parts, different in character. At the beginning we show and shortly discuss - in an informal manner - selected problems in which matrices 'work'. The second part is quite different: it is much more formalized. The examples we describe in that segment are formulated in the mathematical language. Intentionally, we have chosen elementary facts taken from standard programmes of 'math' for students of economics. According to the plan, we collect them and place under unified label 'Matrices'. We also have announced some themes which will be considered in the following articles of the series.
2

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

100%
Rybicki W.
Mathematical Economics
|
2011
|
issue 7 (14)
173-197
EN
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”.
3

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

100%
Rybicki W.
Mathematical Economics
|
2010
|
issue 6(13)
103-126
EN
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).
4

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

100%
Rybicki W.
Mathematical Economics
|
2011
|
issue 7 (14)
169-172
XX
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.
5

ON THE WAYS OF FORMALIZATION AND INTERPRETATION OF THE NOTION OF 'EFFICIENCY' – INTRODUCTORY REMARKS AND SOME EXAMPLES (O sposobach formalizacji i interpretacji pojecia efektywnosci – wstepne rozwazania i niektore przyklady)

100%
Rybicki W.
Mathematical Economics
|
2010
|
issue 6(13)
75-102
EN
In the paper we consider selected formal models, coming from the field of 'pure' mathematics, as well as from some related areas, in which the notion 'efficiency' appears. The presented essay may be seen as a continuation, development and (at the same time) specification of same ideas discussed in the previous article of the author: On manysideness, relativity and complexity of the 'efficiency' (as a category) (in Polish: O wielostronnosci, relatywizmie i zlozonosci kategorii efektywnosci). In addition to the proposals formulated in the above cited paper (concerning the classification and explanation of various 'kinds' of efficiency) we introduce some new ways of meaning of this term, which we suggest to call: (a) basis-type efficiency, (b) sup (inf)-type efficiency, we also define and shortly discuss the following three types of efficiency, related to partial (pre)orders and formal logics, (c) informative capacity (reflecting the 'richness' of an information contained in formulas defining given order), (d) linear similarity - efficiency (expressing a 'distance of the (pre)order from the linear part' of the order in mind), (e) logical efficiency. In the final part we put together (and compare) 'official' terms denoting 'efficiency' and related notions presently functioning in economics, management and praxeology. The further forms of the meaning of notion 'efficiency' are discussed in the 'twin' paper submitted for publication in the present issue of Mathematical Economics.
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