Synopses & Reviews
The goal of this publication is to provide basic tools of differential topology to study systems of nonlinear equations, and to apply them to the analysis of general equilibrium models with complete and incomplete markets. The main content of general equilibrium analysis is to study existence, (local) uniqueness and efficiency of equilibria. To study existence Differential Topology and General Equilibrium with Complete and Incomplete Markets combines two features. As a first step, first order conditions (of agents' maximization problems) and market clearing conditions, instead of aggregate excess demand functions, are used. As a second step, a homotopy argument, stated and proved in relatively elementary manner, is applied to that extended systemof equations. Local uniqueness and smooth dependence of the endogenous variables from the exogenous ones are studied using a version of a so called parametric transversality theorem. In a standard general equilibrium model, all equilibria are efficient, but that is not the case if some imperfection, like incomplete markets, asymmetric information, strategic interaction, is added. Then, for almost all economies, equilibria are inefficient, and an outside institution can Pareto improve upon the market outcome. Those results are proved showing that a well-chosen system of equations has no solutions.
The target audience of Differential Topology and General Equilibrium with Complete and Incomplete Markets consists of researchers interested in economic theory. The needed background is multivariate analysis, basic linear algebra and basic general topology.
Review
From the reviews: "They [authors] should be congratulated for their effort to present this apparatus in Part I of their book in a way that is as simple and transparent as possible, but without making compromises with regard to the required level of generality. I am not aware of any other text in economics that gives a comparable treatment. The book is required reading for anyone that would like to go beyond the traditional complete markets general equilibrium model, and that wants to have a deeper undestanding of the role played by financial markets. [...] it offers a complete account of the subject of incomplete markets, and I would therefore like to recommend it highly." (P.J.J. Herings, Maastricht University in Journal of Economics / Zeitschrift für Nationalökonomie, 81:3 (2004) "Although the book is written, as the authors say, for graduate students in an economics program and stops before really entering the core of differential topology, it is also interesting and profitable for mathematicians being involved with modern theoretical economic problems or applications of differential topology." (Alfred Göpfert, Zentralblatt MATH, Vol. 1103 (5), 2007)
Review
From the reviews:
"They [authors] should be congratulated for their effort to present this apparatus in Part I of their book in a way that is as simple and transparent as possible, but without making compromises with regard to the required level of generality. I am not aware of any other text in economics that gives a comparable treatment. The book is required reading for anyone that would like to go beyond the traditional complete markets general equilibrium model, and that wants to have a deeper undestanding of the role played by financial markets. [...] it offers a complete account of the subject of incomplete markets, and I would therefore like to recommend it highly." (P.J.J. Herings, Maastricht University in Journal of Economics / Zeitschrift für Nationalökonomie, 81:3 (2004)
"Although the book is written, as the authors say, for graduate students in an economics program and stops before really entering the core of differential topology, it is also interesting and profitable for mathematicians being involved with modern theoretical economic problems or applications of differential topology." (Alfred Göpfert, Zentralblatt MATH, Vol. 1103 (5), 2007)
Synopsis
The goal of this publication is to provide basic tools of differential topology to study systems of nonlinear equations, and to apply them to the analysis of general equilibrium models with complete and incomplete markets. The main content of general equilibrium analysis is to study existence, (local) uniqueness and efficiency of equilibria. To study existence Differential Topology and General Equilibrium with Complete and Incomplete Markets combines two features. As a first step, first order conditions (of agents' maximization problems) and market clearing conditions, instead of aggregate excess demand functions, are used. As a second step, a homotopy argument, stated and proved in relatively elementary manner, is applied to that "extended systemof equations. Local uniqueness and smooth dependence of the endogenous variables from the exogenous ones are studied using a version of a so called parametric transversality theorem. In a standard general equilibrium model, all equilibria are efficient, but that is not the case if some imperfection, like incomplete markets, asymmetric information, strategic interaction, is added. Then, for almost all economies, equilibria are inefficient, and an outside institution can Pareto improve upon the market outcome. Those results are proved showing that a well-chosen system of equations has no solutions. The target audience of Differential Topology and General Equilibrium with Complete and Incomplete Markets consists of researchers interested in economic theory. The needed background is multivariate analysis, basic linear algebra and basic general topology.
Table of Contents
List of Figures. Acknowledgments. Introduction.
Part I. 1. Prerequisites.
2. Manifolds in Euclidean Spaces.
3. Differentials.
4. Regular Values.
5. Manifolds with Boundary.
6. Sard's Theorem and Transversality.
7. Homotopy and Degree Theory.
Part II. 8. Exchange Economies.
9. Production Economies.
10. Time, Uncertainty and Incomplete Markets.
11. Numeraire Assets.
12. Nominal Assets.
13. Real Assets.
14. Restricted Participation.
15. Planner Intervention on the Market Outcome. Index.