Mathematical Methods for Economic Analysis - uni-bonn.de

1 Mathematical Methodsfor Economic Analysis Paul SchweinzerSchool of Economics, Statistics and MathematicsBirkbeck College, University of London7-15 Gresse Street, London W1T 1LL, , Fax: This version(9th March 2004) is preliminary and incomplete; I am grateful for corrections or Static analysis91 Mathematical Lagrange theory .. Non-technical Lagrangian .. General Lagrangian .. Jargon & remarks .. Motivation for the rest of the course .. 182 Sets and mappings .. Sets .. Algebraic structures .. Relations .. Extrema and bounds.

2 Mappings .. Elementary combinatorics .. Spaces .. Geometric properties of spaces .. Topological properties of spaces .. Properties of functions .. Continuity .. Differentiability .. Integrability .. Convexity, concavity .. Other properties .. Linear functions onRn.. Linear dependence .. Determinants .. Eigenvectors and eigenvalues .. Quadratic forms .. 523 First applications of Separating hyperplanes .. The irrationality of playing strictly dominated strategies .. The Envelope Theorem.

3 A first statement .. A general statement .. Applications of the Envelope Theorem .. Cost functions .. Parameterised maximisation problems .. Expenditure minimisation and Shephard s Lemma .. The Hicks-Slutsky equation .. The Indirect utility function and Roy s Identity .. Profit functions and Hotelling s Lemma .. 694 Kuhn-Tucker Theory .. The Lagrangian .. The extension proposed by Kuhn and Tucker .. A cookbook approach .. The cookbook version of the Lagrange method .. The cookbook version of the Kuhn-Tucker method.

4 A first cookbook example .. Another cookbook example .. A last cookbook example .. Duality of linear programs .. 855 Measure, probability, and expected Measure .. Measurable sets .. Integrals and measurable functions .. Probability .. Expected utility .. 1026 Machine-supported The programs .. Constrained optimisation problems .. 1187 Fixed Motivation .. Some topological ideas .. Some more details .. Some important definitions .. Existence Theorems .. Brouwer s Theorem .. Kakutani s Theorem.

5 Application: Existence of Nash equilibria .. Tarski s Theorem .. Supermodularity .. 136 CONTENTS5II Dynamic analysis1438 Introduction to dynamic Elements of the theory of ordinary differential equations .. Existence of the solution .. First-order linear ODEs .. First-order non-linear ODEs .. Stability and Phase diagrams .. Higher-order ODEs .. Elements of the theory of ordinary difference equations .. First-order (linear) .. Second-order (linear) O Es .. Stability .. 1629 Introduction to the calculus of Discounting.

6 Depreciation .. Calculus of variation: Derivation of the Euler equation.. Solving the Euler equation .. Transversality condition .. Infinite horizon problems .. 17410 Introduction to discrete Dynamic Assumptions .. Definitions .. The problem statement .. The Bellman equation .. 17911 Deterministic optimal control in continuous Theory I .. Theory II .. Example I: Deterministic optimal control in the Ramseymodel .. Example II: Extending the first Ramsey example .. Example III: Centralised / decentralised equivalenceresults.

7 The command optimum .. The decentralised optimum .. 19612 Stochastic optimal control in continuous Theory III .. Stochastic Calculus .. Theory IV .. Example of stochastic optimal control .. Theory V .. Conclusion .. 216A Appendix217B Some useful results2276 CONTENTSC Notational conventions235 OverviewWe will start with a refresher on linear programming, particularly Lagrange Theory. The problems wewill encounter should provide the motivation for the rest ofthe first part of this course where we willbe concerned mainly with the Mathematical foundations of optimisation theory.

8 This includes a revisionof basic set theory, a look at functions, their continuity and their maximisation inn-dimensional vectorspace (we will only occasionally glimpse beyond finite spaces). The main results are conceptual, that is,not illustrated with numerical computation but composed ofideas that should be helpful to understand avariety of key concepts in modern Microeconomics and Game Theory. We will look at two such results indetail both illustrating concepts from Game Theory: (i) that it is not rational to play a strictly dominatedstrategy and (ii) Nash s equilibrium existence theorem.

9 We will not do many proofs throughout the coursebut those we will do, we will do thoroughly and you will be asked to proof similar results in the course should provide you with the Mathematical tools you will need to follow a master s levelcourse in Economic theory. Familiarity with the material presented in a September course on the levelof Chiang (1984) or Simon and Blume (1994) is assumed and is sufficient to follow the exposition. Thejustification for developing the theory in a rigourous way isto get used to the precise mathematicalnotation that is used in both the journal literature and modern textbooks.

10 We will seek to illustrate theabstract concepts we introduce with Economic examples but this will not always be possible as definitionsare necessarily abstract. More readily applicable material will follow in later sessions. Some sections areflagged with daggers indicating that they can be skipped on first main textbook we will use for the Autumn term is (Sundaram1996). It is more technical and to anextent more difficult than the course itself. We will cover about a third of the book. If you are interestedin formal Analysis or are planning to further pursue Economic research, I strongly encourage you to workthrough this text.