Warsaw Fall Seminars In Mathematical Economics

To write everything about nothing, or to write nothing about everything: this is the problem. (Anonym, circa 1996-97) The first idea to write a book on M athematical Economics, more or less ordered in a historical sequence, occurred to me in 1995, when I was asked, by Istituto delta Enciclopedia Italiana, to write the entry "Storia dell'economia 1 2 matematica" , for the collective work "Storia deI XX Secolo". I thought that it would be interesting to elaborate on the text presented to the editors, to turn it into a book aiming at giving a panorama of what, in my opinion, are the main 20th century contributions to mathematical eco nomics. Of course, only a narrow set of the contributions made by economic theorists could be included, both for space limitations and necessity, because 3 of the limited competence of any single author. For instance, I have paid very limited attention to what is now called Macroeconomics, and also to Game Theory, which actually has grown so much as to acquire scientific in dependence as a living branch of applied mathematics. For the same reason, I have also left completely untouched such fields as Mathematical Finance, Public Economics, Theory of Taxation, etc. I have always based my presentation on published material only, assuming that what is contained in working papers still waits to be confirmed, possibly in the first years of the 21th century.

On February 20, 1978, the Department of Econometrics of the University of Tilburg organized a symposium on Convex Analysis and Mathematical th Economics to commemorate the 50 anniversary of the University. The general theme of the anniversary celebration was "innovation" and since an important part of the departments' theoretical work is con centrated on mathematical economics, the above mentioned theme was chosen. The scientific part of the Symposium consisted of four lectures, three of them are included in an adapted form in this volume, the fourth lec ture was a mathematical one with the title "On the development of the application of convexity". The three papers included concern recent developments in the relations between convex analysis and mathematical economics. Dr. P.H.M. Ruys and Dr. H.N. Weddepohl (University of Tilburg) study in their paper "Economic theory and duality", the relations between optimality and equilibrium concepts in economic theory and various duality concepts in convex analysis. The models are introduced with an individual facing a decision in an optimization problem. Next, an n person decision problem is analyzed, and the following concepts are defined: optimum, relative optimum, Nash-equilibrium, and Pareto-optimum.

Since the beginning of the fifties, the ruling paradigm in the discipline of economics has been that of a competitive general equilibrium. Associated dynamic analyses have therefore been preoccupied with the stability of this equilibrium state, corresponding simply to studies of comparative statics. The need to permeate the boundaries of this paradigm in order to open up new pathways for genuine dynamic analysis is now pressing. The contributions contained in this volume spring from this very ambition. A growing circle of economists have recently been inspired by two distinct but complementary sources: (i) the pathbreaking work of Joseph Schumpeter, and (ii) recent contributions to physics, chemistry and theoretical biology. It turns out that problems which are firmly rooted in the economic discipline, such as innovation, technological change, business cycles and economic development, contain many clear parallels with phenomena from the natural sciences such as the slaving principle, adiabatic elimination and self-organization. In such dynamic worlds, adjustment processes and adaptive behaviour are modelled with the aid of the mathematical theory of nonlinear dynamical systems. The dynamics is defined for a much wider set of conditions or states than simply a set of competitive equilibria. A common objective is to study and classify ways in which the qualitative properties of each system change as the parameters describing the system vary.

The seminar for which the proceedings are published here evolved from a cooperative research program on bilinear systems and applications to immunology at the Oregon State University and at the University of Rome. The topics include more general forms of variable structure systems which may be divided into categories of mathematical system theory, economic applications and biological applications. Throughout the seminar there was emphasis on the integration of theory and app- cation. In most cases, theoretical derivations are motivated by their need to solve practical problems. In reading the proceedings, it becomes apparent that bilinear systems, quadratic systems and more general variable structure or adaptive systems become natural models in many cases and excellent approximations in others. It is seen that linear systems have very limited use particularly in economics and biology. Variable structure systems are analyzed in terms of structure,volterra kernels, system modelling, parameter identification,- controllability and Lie algebra to mention a few. Certainly, it ia nbt possible to present a complete tre- ment of these numerous topics, but at the Same time the unifying power of the systems approach and variable structure systems is shown.

No detailed description available for "Proceedings of the Bakuriani Colloquium in Honour of Yu.V. Prohorov, Bakuriani, Georgia, USSR, 24 February-4 March, 1990".