Optima And Equilibria

Progress in the theory of economic equilibria and in game theory has proceeded hand in hand with that of the mathematical tools used in the field, namely nonlinear analysis and, in particular, convex analysis. Jean-Pierre Aubin, one of the leading specialists in nonlinear analysis and its application to economics, has written a rigorous and concise - yet still elementary and self-contained - textbook providing the mathematical tools needed to study optima and equilibria, as solutions to problems, arising in economics, management sciences, operations research, cooperative and non-cooperative games, fuzzy games etc. It begins with the foundations of optimization theory, and mathematical programming, and in particular convex and nonsmooth analysis. Nonlinear analysis is then presented, first game-theoretically, then in the framework of set valued analysis. These results are then applied to the main classes of economic equilibria. The book contains numerous exercises and problems: the latter allow the reader to venture into areas of nonlinear analysis that lie beyond the scope of the book and of most graduate courses.

By Way of Warning As in ordinary language, metaphors may be used in mathematics to explain a given phenomenon by associating it with another which is ( or is considered to be) more familiar. It is this sense of familiarity, whether individual or collective, innate or acquired by education, which enables one to convince oneself that one has understood the phenomenon in question. Contrary to popular opinion, mathematics is not simply a richer or more precise language. Mathematical reasoning is a separate faculty possessed by all human brains, just like the ability to compose or listen to music, to paint or look at paintings, to believe in and follow cultural or moral codes, etc. But it is impossible (and dangerous) to compare these various faculties within a hierarchical framework; in particular, one cannot speak of the superiority of the language of mathematics. Naturally, the construction of mathematical metaphors requires the autonomous development of the discipline to provide theories which may be substituted for or associated with the phenomena to be explained. This is the domain of pure mathematics. The construction of the mathematical corpus obeys its own logic, like that of literature, music or art. In all these domains, an aesthetic satisfaction is at once the objective of the creative activity and a signal which enables one to recognise successful works. (Likewise, in all these domains, fashionable phenomena - reflecting social consensus - are used to develop aesthetic criteria).

This book presents an original exposition of general equilibrium theory for advanced undergraduate and graduate-level students of economics. It contains detailed discussions of economic efficiency, competitive equilibrium, the first and second welfare theorems, the Kuhn-Tucker approach to general equilibrium, the Arrow-Debreu model, and rational expectations equilibrium and the permanent income hypothesis. Truman Bewley also treats optimal growth and overlapping generations models as special cases of the general equilibrium model. He uses the model and the first and second welfare theorems to explain the main ideas of insurance, capital theory, growth theory, and social security. It enables him to present a unified approach to portions of macro- as well as microeconomic theory. The book contains problems sets for most chapters.

The first of two volumes, this edited proceedings book features research presented at the XVI International Conference on Hyperbolic Problems held in Aachen, Germany in summer 2016. It focuses on the theoretical, applied, and computational aspects of hyperbolic partial differential equations (systems of hyperbolic conservation laws, wave equations, etc.) and of related mathematical models (PDEs of mixed type, kinetic equations, nonlocal or/and discrete models) found in the field of applied sciences.

Dean Corbae, Maxwell B.

This book constitutes the refereed conference proceedings of the 27th International Colloquium on Structural Information and Communication Complexity, SIROCCO 2020, held in Paderborn, Germany, in June 2020.* The 19 full papers and 2 invited papers presented in this book were carefully reviewed and selected from 41 submissions. They are divided into seven sections, i.e. invited papers; mobile robots; dynamic graphs; network communication; multi-agent systems; communication complexity; and game theory. * The conference was held virtually due to the COVID-19 pandemic.