Mathematical Methods for Economic Analysis∗

244 Pages·2004·1.6 MB·English

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: [phone], Fax: [phone]
∗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
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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