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SYLLABUS FOR QUANTITATIVE METHODS

Class 4. Determinants and the Matrix Inverse   39

4.1 Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.1.1 The volume of a matrix . . . . . . . . . . . . . . . . . . . . . 40
4.1.2 Determinants of small matrices . . . . . . . . . . . . . . . . . 41

4.2 Properties of determinants . . . . . . . . . . . . . . . . . . . . . . . . 41

4.3 The matrix inverse . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.3.1 Construction of the inverse . . . . . . . . . . . . . . . . . . . . 45
4.3.2 Properties of inverses . . . . . . . . . . . . . . . . . . . . . . . 46
4.3.3 Elementary decomposition . . . . . . . . . . . . . . . . . . . . 47
4.4 The adjoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.4.1 Cramer’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

Class 5. Eigenpairs   51

5.1 Eigenvalues and eigenvectors . . . . . . . . . . . . . . . . . . . . . . . 51
5.1.1 The characteristic polynomial . . . . . . . . . . . . . . . . . . 51
Application: Markov chains . . . . . . . . . . . . . . . . . . . . . . . 53
Application: Google’s PageRank . . . . . . . . . . . . . . . . . . . . . 53
5.1.2 Properties of eigenvalues and eigenvectors . . . . . . . . . . . 54

5.2 Spectral decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5.2.1 Diagonalization . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5.2.2 Uncoupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
A metaphor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5.3 Quadratic forms and sign definiteness . . . . . . . . . . . . . . . . . . 59
5.3.1 Tests for sign definiteness . . . . . . . . . . . . . . . . . . . . 59

III BASIC TOPOLOGY   61

Class 6. Topological Set Properties   61

6.1 Distance measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

6.2 Openness and closedness . . . . . . . . . . . . . . . . . . . . . . . . . 62
6.2.1 Open balls and open sets . . . . . . . . . . . . . . . . . . . . . 62
6.2.2 Relative openness . . . . . . . . . . . . . . . . . . . . . . . . . 64
6.2.3 Unions and intersections of open sets . . . . . . . . . . . . . . 64
6.2.4 Closed sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
6.2.5 Interior, closure, and boundary of a set . . . . . . . . . . . . . 66

6.3 Compactness and connectedness . . . . . . . . . . . . . . . . . . . . . 67
6.3.1 Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
6.3.2 Connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

Class 7. Limits and Convergence 71

7.1 Limit points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

7.2 Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
7.2.1 Convergence of sequences . . . . . . . . . . . . . . . . . . . . 73
7.2.2 Cauchy sequences . . . . . . . . . . . . . . . . . . . . . . . . . 74

7.3 The Bolzano-Weierstrass Theorem . . . . . . . . . . . . . . . . . . . . 75

7.4 Functions and their limits . . . . . . . . . . . . . . . . . . . . . . . . 77
7.4.1 Properties of functions . . . . . . . . . . . . . . . . . . . . . . 77
7.4.2 Function limits . . . . . . . . . . . . . . . . . . . . . . . . . . 78

Class 8. Continuity 81

8.1 Continuous functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
8.1.1 Continuity and function limits . . . . . . . . . . . . . . . . . . 81
8.1.2 Continuity and sequence limits . . . . . . . . . . . . . . . . . 82

8.2 Continuity and open sets . . . . . . . . . . . . . . . . . . . . . . . . . 83
8.2.1 Image and inverse image . . . . . . . . . . . . . . . . . . . . . 83
8.2.2 An alternative definition of continuity . . . . . . . . . . . . . . 84
8.2.3 Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

8.3 The Weierstrass Theorem . . . . . . . . . . . . . . . . . . . . . . . . 86

8.4 Semi-continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

IV CALCULUS   91

Class 9. Differentiation   91

9.1 Differentiability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
9.1.1 Some rules of differentiation . . . . . . . . . . . . . . . . . . . 92
9.1.2 Differentiability and continuity . . . . . . . . . . . . . . . . . 94
9.1.3 Elasticities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

9.2 Linear approximations and applications . . . . . . . . . . . . . . . . . 96
9.2.1 Tangents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
9.2.2 Chain rule, product rule , quotient rule . . . . . . . . . . . . . 98
9.2.3 L’Hˆopital’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . 99

9.3 Differentiation of implicit and inverse functions . . . . . . . . . . . . 100
9.3.1 Implicit functions . . . . . . . . . . . . . . . . . . . . . . . . . 100
9.3.2 Implicit differentiation and differentials . . . . . . . . . . . . . 101
9.3.3 Inverse functions . . . . . . . . . . . . . . . . . . . . . . . . . 103

9.4 The Intermediate Value Theorem . . . . . . . . . . . . . . . . . . . . 104

Class 10. Multivariate Calculus 105

10.1 Functions of several variables . . . . . . . . . . . . . . . . . . . . . . 105
10.1.1 Partial derivatives . . . . . . . . . . . . . . . . . . . . . . . . . 105
10.1.2 The Jacobian derivative . . . . . . . . . . . . . . . . . . . . . 106

10.2 Linear approximations and applications . . . . . . . . . . . . . . . . . 107
10.2.1 Tangent plane of a graph . . . . . . . . . . . . . . . . . . . . . 107
10.2.2 The chain rule in general . . . . . . . . . . . . . . . . . . . . . 108
10.2.3 Directional derivatives . . . . . . . . . . . . . . . . . . . . . . 110

10.3 Implicit functions of several variables . . . . . . . . . . . . . . . . . . 112
10.3.1 An example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
10.3.2 The Implicit Function Theorem . . . . . . . . . . . . . . . . . 114
10.3.3 Level curves and gradients . . . . . . . . . . . . . . . . . . . . 114
Application: Comparative statics . . . . . . . . . . . . . . . . . . . . 116

Class 11. Higher- Order Derivatives 119

11.1 Functions of one variable . . . . . . . . . . . . . . . . . . . . . . . . . 119
11.1.1 Concavity and convexity . . . . . . . . . . . . . . . . . . . . . 120

11.2 Taylor expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
11.2.1 Quadratic approximations . . . . . . . . . . . . . . . . . . . . 122
11.2.2 Taylor’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 123
11.2.3 Taylor series . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

11.3 Functions of several variables . . . . . . . . . . . . . . . . . . . . . . 125
11.3.1 The Hessian . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
11.3.2 Young’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 125
11.3.3 Quadratic approximations of multi- variable functions . . . . . 127
11.3.4 Concavity and convexity in several variables . . . . . . . . . . 128

Class 12. Integration and Differential Equations 131

12.1 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
12.1.1 The Riemann integral . . . . . . . . . . . . . . . . . . . . . . 131
12.1.2 Antiderivatives . . . . . . . . . . . . . . . . . . . . . . . . . . 133
12.1.3 The Fundamental Theorems of Calculus . . . . . . . . . . . . 134
12.1.4 Rules of integration . . . . . . . . . . . . . . . . . . . . . . . . 135
12.1.5 Integration by parts . . . . . . . . . . . . . . . . . . . . . . . 135

12.2 Integration in higher dimensions . . . . . . . . . . . . . . . . . . . . . 136
12.2.1 The rectangular case . . . . . . . . . . . . . . . . . . . . . . . 136
12.2.2 The non-rectangular case . . . . . . . . . . . . . . . . . . . . . 138

12.3 Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
12.3.1 Homogenous linear first -order differential equations . . . . . . 140
12.3.2 Non-homogenous differential equations . . . . . . . . . . . . . 140

V OPTIMIZATION THEORY   143
Class 13. Unconstrained Optimization   143

13.1 Maxima and minima . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
13.1.1 Global extrema . . . . . . . . . . . . . . . . . . . . . . . . . . 143
13.1.2 Local extrema . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

13.2 Finding interior extrema . . . . . . . . . . . . . . . . . . . . . . . . . 145
13.2.1 Necessary conditions . . . . . . . . . . . . . . . . . . . . . . . 145
13.2.2 Sufficient conditions . . . . . . . . . . . . . . . . . . . . . . . 146
13.2.3 The nth-derivative test . . . . . . . . . . . . . . . . . . . . . . 147

13.3 The Envelope Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 149

Class 14. Constrained Optimization 153

14.1 Optimization subject to an equality constraint . . . . . . . . . . . . . 153
14.1.1 Translation into an unconstrained problem . . . . . . . . . . . 153
14.1.2 The tangency condition . . . . . . . . . . . . . . . . . . . . . 155
14.1.3 The Lagrangean . . . . . . . . . . . . . . . . . . . . . . . . . . 157

14.2 Optimization subject to an inequality constraint . . . . . . . . . . . . 158
14.2.1 Conditions for an optimum . . . . . . . . . . . . . . . . . . . . 158
14.2.2 The Kuhn-Tucker Theorem . . . . . . . . . . . . . . . . . . . 160
14.2.3 Interpretation of the multiplier . . . . . . . . . . . . . . . . . 162
Application: Profit maximum subject to a technological constraint . . 163

14.3 Multiple constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

Class 15. Dynamic Optimization   167

15.1 Problem setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
15.1.1 Intertemporal constraints . . . . . . . . . . . . . . . . . . . . . 167
15.1.2 Continuous time . . . . . . . . . . . . . . . . . . . . . . . . . 168
15.1.3 Example: The consumption-savings problem . . . . . . . . . . 169

15.2 Optimal control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
15.2.1 Conditions for a dynamic optimum . . . . . . . . . . . . . . . 171
15.2.2 The Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . 172
15.2.3 Intertemporal arbitrage and the Maximum Principle . . . . . . 173
Application: Life cycle savings profile . . . . . . . . . . . . . . . . . . 174

15.3 Dynamic programming . . . . . . . . . . . . . . . . . . . . . . . . . . 176
15.3.1 The Bellman equation . . . . . . . . . . . . . . . . . . . . . . 176
15.3.2 The Principle of Optimality . . . . . . . . . . . . . . . . . . . 177

Index   179

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