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Mathematical Methods for Economic Analysis ∗ Paul Schweinzer School of Economics, Statistics and Mathematics Birkbeck College, University of London 7-15 Gresse Street, London W1T 1LL, UK Email: [email protected] Tel: 020-7631.6445, Fax: 020-7631.6416 ∗Thisversion(9th March 2004) is preliminary and incomplete; I am grateful for corrections or suggestions. 2 Contents I Static analysis 9 1 Mathematical programming 11 1.1 Lagrange theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.1.1 Non-technical Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.1.2 General Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.1.3 Jargon & remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.2 Motivation for the rest of the course . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2 Fundamentals 21 2.1 Sets and mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.1.1 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.2 Algebraic structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.2.1 Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.2.2 Extrema and bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.2.3 Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.3 Elementary combinatorics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.4 Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.4.1 Geometric properties of spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.4.2 Topological properties of spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.5 Properties of functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.5.1 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.5.2 Differentiability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.5.3 Integrability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.5.4 Convexity, concavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 2.5.5 Other properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 n 2.6 Linear functions on R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.6.1 Linear dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.6.2 Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 2.6.3 Eigenvectors and eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 2.6.4 Quadratic forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3 First applications of fundamentals 55 3.1 Separating hyperplanes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.2 The irrationality of playing strictly dominated strategies . . . . . . . . . . . . . . . . . . . . 57 3.3 The Envelope Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.3.1 A first statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3 4 CONTENTS 3.3.2 A general statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.4 Applications of the Envelope Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.4.1 Cost functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.4.2 Parameterisedmaximisation problems . . . . . . . . . . . . . . . . . . . . . . . . . . 67 3.4.3 Expenditure minimisation and Shephard’s Lemma . . . . . . . . . . . . . . . . . . . 68 3.4.4 The Hicks-Slutsky equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.4.5 The Indirect utility function and Roy’s Identity . . . . . . . . . . . . . . . . . . . . . 69 3.4.6 Profit functions and Hotelling’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . 69 4 Kuhn-Tucker theory 73 4.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4.1.1 The Lagrangian. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4.1.2 The extension proposed by Kuhn and Tucker . . . . . . . . . . . . . . . . . . . . . . 76 4.2 A cookbook approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 4.2.1 The cookbook version of the Lagrange method . . . . . . . . . . . . . . . . . . . . . 79 4.2.2 The cookbook version of the Kuhn-Tucker method . . . . . . . . . . . . . . . . . . . 79 4.2.3 A first cookbook example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 4.2.4 Another cookbook example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.2.5 A last cookbook example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.3 Duality of linear programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 5 Measure, probability, and expected utility 89 5.1 Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 5.1.1 Measurable sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 5.1.2 Integrals and measurable functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 5.2 Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 5.3 Expected utility. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 6 Machine-supported mathematics 113 6.1 The programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 6.2 Constrained optimisation problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 7 Fixed points 121 7.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 7.1.1 Some topological ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 7.1.2 Some more details† . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 7.1.3 Some important definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 7.2 Existence Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 7.2.1 Brouwer’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 7.2.2 Kakutani’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 7.2.3 Application: Existence of Nash equilibria . . . . . . . . . . . . . . . . . . . . . . . . 131 7.2.4 Tarski’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 7.2.5 Supermodularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 CONTENTS 5 II Dynamic analysis 143 8 Introduction to dynamic systems 145 8.1 Elements of the theory of ordinary differential equations . . . . . . . . . . . . . . . . . . . . 146 8.1.1 Existence of the solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 8.1.2 First-order linear ODEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 8.1.3 First-order non-linear ODEs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 8.1.4 Stability and Phase diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 8.1.5 Higher-order ODEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 8.2 Elements of the theory of ordinary difference equations . . . . . . . . . . . . . . . . . . . . . 157 8.2.1 First-order (linear) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 8.2.2 Second-order (linear) O∆Es . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 8.2.3 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 9 Introduction to the calculus of variation 165 9.1 Discounting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 9.2 Depreciation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 9.3 Calculus of variation: Derivation of the Euler equation . . . . . . . . . . . . . . . . . . . . . 167 9.4 Solving the Euler equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 9.5 Transversalitycondition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 9.6 Infinite horizon problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 10 Introduction to discrete Dynamic Programming 177 10.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 10.2 Definitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 10.3 The problem statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 10.4 The Bellman equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 11 Deterministic optimal control in continuous time 181 11.1 Theory I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 11.2 Theory II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 11.3 Example I: Deterministic optimal control in the Ramsey model . . . . . . . . . . . . . . . . 189 11.4 Example II: Extending the first Ramsey example‡ . . . . . . . . . . . . . . . . . . . . . . . 193 11.5 Example III: Centralised / decentralised equivalence results‡ . . . . . . . . . . . . . . . . . . 194 11.5.1 The command optimum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 11.5.2 The decentralised optimum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 12 Stochastic optimal control in continuous time 203 12.1 Theory III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 12.2 Stochastic Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 12.3 Theory IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 12.4 Example of stochastic optimal control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 12.5 Theory V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 12.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 A Appendix 217 B Some useful results 227 6 CONTENTS C Notational conventions 235 Overview We will start with a refresher on linear programming, particularly Lagrange Theory. The problems we will encounter should provide the motivation for the rest of the first part of this course where we will be concerned mainly with the mathematical foundations of optimisation theory. This includes a revision of basic set theory, a look at functions, their continuity and their maximisation in n-dimensional vector space (we will only occasionally glimpse beyond finite spaces). The main results are conceptual, that is, not illustrated with numerical computation but composed of ideas that should be helpful to understand a variety of key concepts in modern Microeconomics and Game Theory. We will look at two such results in detail—bothillustratingconceptsfromGameTheory:(i)thatitisnotrationaltoplayastrictlydominated strategyand(ii)Nash’sequilibriumexistencetheorem. Wewillnotdomanyproofsthroughoutthecourse but those we will do, we will do thoroughly and you will be askedto proof similar results in the exercises. The course should provide you with the mathematical tools you will need to follow a master’s level course in economic theory. Familiarity with the material presented in a ‘September course’ on the level of Chiang (1984) or Simon and Blume (1994) is assumed and is sufficient to follow the exposition. The justification for developing the theory in a rigourous way is to get used to the precise mathematical notation that is used in both the journal literature and modern textbooks. We will seek to illustrate the abstract concepts we introduce with economic examples but this will not alwaysbe possible as definitions are necessarily abstract. More readily applicable material will follow in later sessions. Some sections are flagged with daggers† indicating that they can be skipped on first reading. ThemaintextbookwewillusefortheAutumntermis(Sundaram1996). Itismoretechnicalandtoan extent more difficult than the course itself. We will cover about a third of the book. If you are interested in formal analysis or are planning to further pursue economic research, I strongly encourage you to work through this text. If you find yourself struggling, consult a suitable text from the reference section. The second part of the course (starting in December) will be devoted to the main optimisation tool used in dynamic settings as in most modern Macroeconomics: Dynamic Control Theory. We will focus on the Bellman approach and develop the Hamiltonian in both a deterministic and stochastic setting. In addition we will derive a cookbook-style recipe of how to solve the optimisation problems you will face in the Macro-part of your economic theory lectures. To get a firm grasp of this you will need most of the fundamentals we introduced in the Autumn term sessions. The main text we will use in the Spring term is (Obstfeld 1992); it forms the basis of section (11) and almost all of (12). You should supplement your reading by reference to (Kamien and Schwartz 1991) which, although a very good book, is not exactly cheap. Since we will not touch upon more than a quarter of the text you should only buy it if you lost your heart to dynamic optimisation. Likeineverymathematicscourse:Unless youalreadyknowthe materialcoveredquite well,there isno wayyoucanunderstandwhatisgoingonwithoutdoingatleastsomeofthe exercisesindicatedatthe end of each section. Let me close with a word of warning: This is only the third time these notes are used for teaching. This means that numerous mistakes, typos and ambiguities have been removed by the students using the notes in previous years. I am most grateful to them—but I assure you that there are enough remaining. I apologise for these and would be happy about comments and suggestions. I hope we will have an interesting and stimulating course. Paul Schweinzer, Summer 2002.1 1IamgratefultoJohnHillas,UniversityofAuckland,forallowingmetousepartofhislecturenotesfortheintroductory sectionontheLagrangian. IoweasimilardebttoMauriceObstfeld,UniversityofCaliforniaatBerkeley,forallowingmeto incorporatehispaperintothespring-termpartofthenotes. PedroBacao—endowedwithendlessenergyandpatience—read the whole draft and provided very helpful comments and suggestions and an anonymous referee contributed most detailed notes andcorrections. 7 8 CONTENTS “The good Christian should beware of mathematics and all those who make empty prophecies. The danger already ex- ists that mathematicians have made a covenant with the devil to darken the spirit and to confine man in the bonds of hell.” Saint Augustine (4th C.) Part I Static analysis 9

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