Mathematical Methods in Economics and Social Choice Second Edition

Springer Texts in Business and Economics Schofield Springer Texts in Business and Economics Norman Schofield Mathematical Methods in Economics and Social Choice Second Edition In recent years, the usual optimization techniques, which have proved so useful in microeconomic theory, have been extended to incorporate more powerful topological and differential methods, and these methods have led to new results on the qualitative behavior of general economic and political systems. These developments have neces- sarily resulted in an increase in the degree of formalism in the publications in the academic journals. This formalism can often deter graduate students. The progression of ideas presented in this book will familiarize the student with the geometric concepts Norman Schofield underlying these topological methods, and, as a result, make mathematical economics, general equilibrium theory, and social choice theory more accessible. 1 Mathematical Mathematical Methods in Methods in Mathematical Economics and Social Choice Economics Methods in Economics and Social Choice Second Edition Business / Economics ISBN 978-3-642-39817-9 9 7 8 3 6 4 2 3 9 8 1 7 9 2nd Ed. Dedicated to the memory of Jeffrey Banks 1 and Richard McKelvey 2 UNCORRECTED PROOF UNCORRECTED PROOF Foreword 1 The use of mathematics in the social sciences is expanding both in breadth and 2 depth at an increasing rate. It has made its way from economics into the other social 3 sciences, often accompanied by the same controversy that raged in economics in the 4 1950’s. And its use has deepened from calculus to topology and measure theory to 5 the methods of differential topology and functional analysis. 6 The reasons for this expansion are several. First, and perhaps foremost, math- 7 ematics makes communication between researchers succinct and precise. Second, 8 it helps make assumptions and models clear; this bypasses arguments in the field 9 that are a result of different implicit assumptions. Third, proofs are rigorous, so 10 mathematics helps avoid mistakes in the literature. Fourth, its use often provides 11 more insights into the models. And finally, the models can be applied to different 12 contexts without repeating the analysis, simply by renaming the symbols. 13 Of course, the formulation of social science questions must precede the construc- 14 tion of models and the distillation of these models down to mathematical problems, 15 for otherwise the assumptions might be inappropriate. 16 A consequence of the pervasive use of mathematics in our research is a 17 change in the level of mathematics training required of our graduate students. 18 We need reference and graduate text books that address applications of advanced 19 mathematics to a widening range of social sciences. This book fills that need. 20 Many years ago, Bill Riker introduced me to Norman Schofield’s work and 21 then to Norman. He is unique in his ability to span the social sciences and apply 22 integrative mathematical reasoning to them all. The emphasis on his work and 23 hisUNCORRECTED book is on smooth models and techniques, while the motivating PROOF examples for 24 presentation of the mathematics are drawn primarily from economics and political 25 science. The reader is taken from basic set theory to the mathematics used to solve 26 problems at the cutting edge of research. Students in every social science will 27 find exposure to this mode of analysis useful; it elucidates the common threads 28 in different fields. Speculations at the end of Chapter 5 provide students and 29 researchers with many open research questions related to the content of the first 30 four chapters. The answers are in these chapters, and a goal of the reader should be 31 to write Chapter 6. 32 Foreword Marcus Berliant 33 St. Louis, Missouri, 2002 34 UNCORRECTED PROOF Author’s Preface 1 In recent years, the optimisation techniques, which have proved so useful in microe- 2 conomic theory, have been extended to incorporate more powerful topological and 3 differential methods. These methods have led to new results on the qualitative 4 behaviour of general economic and political systems. However, these developments 5 have also led to an increase in the degree of formalism in published work. This 6 formalism can often deter graduate students. My hope is that the progression of 7 ideas presented in these lecture notes will familiarise the student with the geometric 8 concepts underlying these topological methods, and, as a result, make mathematical 9 economics, general equilibrium theory, and social choice theory more accessible. 10 The first chapter of the book introduces the general idea of mathematical 11 structure and representation, while the second chapter analyses linear systems and 12 the representation of transformations of linear systems by matrices. In the third 13 chapter, topological ideas and continuity are introduced and used to solve convex 14 optimisation problems. These techniques are also used to examine existence of a 15 ”social equilibrium.” Chapter four then goes on to study calculus techniques using 16 a linear approximation, the differential, of a function to study its ”local” behaviour. 17 The book is not intended to cover the full extent of mathematical economics 18 or general equilibrium theory. However, in the last sections of the third and fourth 19 chapters I have introduced some of the standard tools of economic theory, namely 20 the Kuhn Tucker Theorem, together with some elements of convex analysis and 21 procedures using the Langrangian. Chapter four provides examples of consumer and 22 producer optimisation. The final section of the chapter also discusses, in a heuristic 23 fashion,UNCORRECTED the smooth or critical Pareto set and the idea of a regular PROOF economy. The fifth 24 and final chapter is somewhat more advanced, and extends the differential calculus 25 of a real valued function to the analysis of a smooth function between ”local” vector 26 spaces, or manifolds. Modern singularity theory is the study and classification of all 27 such smooth functions, and the purpose of the final chapter to use this perspective to 28 obtain a generic or typical picture of the Pareto set and the set of Walrasian equilibria 29 of an exchange economy. 30 Since the underlying mathematics of this final section are rather difficult, I 31 have not attempted rigorous proofs, but rather have sought to lay out the natural 32 Author’s Preface path of development from elementary differential calculus to the powerful tools of 33 singularity theory. In the text I have referred to work of Debreu, Balasko, Smale, 34 and Saari, among others who, in the last few years, have used the tools of singularity 35 theory to develop a deeper insight into the geometric structure of both the economy 36 and the polity. These ideas are at the heart of recent notions of ”chaos.” Some 37 speculations on this profound way of thinking about the world are offered in section 38 5.6. Review exercises are provided at the end of the book. 39 I thank Annette Milford for typing the manuscript and Diana Ivanov for the 40 preparation of the figures. 41 I am also indebted to my graduate students for the pertinent questions they 42 asked during the courses on mathematical methods in economics and social choice, 43 which I have given at Essex University, the California Institute of Technology, and 44 Washington University in St. Louis. 45 In particular, while I was at the California Institute of Technology I had the 46 privilege of working with Richard McKelvey and of discussing ideas in social choice 47 theory with Jeff Banks. It is a great loss that they have both passed away. This book 48 is dedicated to their memory. 49 Norman Schofield 50 St. Louis, Missouri 51 UNCORRECTED PROOF Contents 1 1 Sets, Relations, and Preferences 2 1.1 Elements of Set Theory 3 1.1.1 A Set Theory 4 1.1.2 A Propositional Calculus 5 1.1.3 Partitions and Covers 6 1.1.4 The Universal and Existential Quantifiers 7 1.2 Relations, Functions and Operations 8 1.2.1 Relations 9 1.2.2 Mappings 10 1.2.3 Function 11 1.3 Groups and Morphisms 12 1.4 Preferences and Choices 13 1.4.1 Preference Relations 14 1.4.2 Rationality 15 1.4.3 Choices 16 1.5 Social Choice and Arrow’s Impossibility Theorem 17 1.5.1 Oligarchies and Filters 18 1.5.2 Acyclicity and The Collegium 19 References 20 2 Linear Spaces and Transformations 21 2.1 Vector Spaces 22 UNCORRECTED2.2 Linear Transformations PROOF23 2.2.1 Matrices 24 2.2.2 The Dimension Theorem 25 2.2.3 The General Linear Group 26 2.2.4 Change of Basis 27 2.2.5 Examples 28 2.3 Canonical Representation 29 2.3.1 Examples 30 2.3.2 Symmetric Matrices and Quadratic Forms 31 Contents 2.3.3 Examples 32 2.4 Geometric Interpretation of a Linear Transformation 33 3 Topology and Convex Optimisation 34 3.1 A Topological Space 35 3.1.1 Scalar Product and Norms 36 3.1.2 A Topology on a Set 37 3.2 Continuity 38 3.3 Compactness 39 3.4 Convexity 40 3.4.1 A Convex Set 41 3.4.2 Examples 42 3.4.3 Separation Properties of Convex Sets 43 3.5 Optimisation on Convex Sets 44 3.5.1 Optimisation of a Convex Preference 45 Correspondence 46 3.6 Kuhn-Tucker Theorem 47 3.7 Choice on Compact Sets 48 3.8 Political and Economic Choice 49 References 50 4 Differential Calculus and Smooth Optimisation 51 4.1 Differential of a Function 52 4.2 C r -Differentiable Functions 53 4.2.1 The Hessian 54 4.2.2 Taylor’s Theorem 55 4.2.3 Critical Points of a Function 56 4.3 Constrained Optimisation 57 4.3.1 Concave and Quasi-concave Functions 58 4.3.2 Economic Optimisation with Exogenous Prices 59 4.4

Load more