List of figures ............................................. ix
List of tables .............................................. xi
Foreword .................................................. xiii
Preface ..................................................... xv
Acknowledgements .......................................... xvii
List of abbreviations .................................... xviii
Notation and preliminaries ................................. xix
Part I MATHEMATICS
Introduction ................................................. 3
1 Systems of linear equations and matrices ..................... 5
1.1 Introduction ............................................ 5
1.2 Linear equations and examples ........................... 5
1.3 Matrix operations ...................................... 11
1.4 Rules of matrix algebra ................................ 14
1.5 Some special types of matrix and associated rules ...... 15
2 Determinants ................................................ 30
2.1 Introduction ........................................... 30
2.2 Preliminaries .......................................... 30
2.3 Definition and properties .............................. 31
2.4 Co-factor expansions of determinants ................... 34
2.5 Solution of systems of equations ....................... 39
3 Eigenvalues and eigenvectors ................................ 53
3.1 Introduction ........................................... 53
3.2 Definitions and illustration ........................... 53
3.3 Computation ............................................ 54
3.4 Unit eigenvalues ....................................... 58
3.5 Similar matrices ....................................... 59
3.6 Diagonalization ........................................ 59
4 Conic sections, quadratic forms and definite matrices ....... 71
4.1 Introduction ........................................... 71
4.2 Conic sections ......................................... 71
4.3 Quadratic forms ........................................ 76
4.4 Definite matrices ...................................... 77
5 Vectors and vector spaces ................................... 88
5.1 Introduction ........................................... 88
5.2 Vectors in 2-space and 3-space ......................... 88
5.3 n-Dimensional Euclidean vector spaces ................. 100
5.4 General vector spaces ................................. 101
6 Linear transformations ..................................... 128
6.1 Introduction .......................................... 128
6.2 Definitions and illustrations ......................... 128
6.3 Properties of linear transformations .................. 131
6.4 Linear transformations from  n to m .................. 137
6.5 Matrices of linear transformations .................... 138
7 Foundations for vector calculus ............................ 143
7.1 Introduction .......................................... 143
7.2 Affine combinations, sets, hulls and functions ........ 143
7.3 Convex combinations, sets, hulls and functions ........ 146
7.4 Subsets of n-dimensional spaces ....................... 148
7.5 Basic topology ........................................ 154
7.6 Supporting and separating hyperplane theorems ......... 157
7.7 Visualizing functions of several variables ............ 158
7.8 Limits and continuity ................................. 159
7.9 Fundamental theorem of calculus ....................... 162
8 Difference equations ....................................... 167
8.1 Introduction .......................................... 167
8.2 Definitions and classifications ....................... 167
8.3 Linear, first-order difference equations .............. 172
8.4 Linear, autonomous, higher-order difference
equations ............................................. 181
8.5 Systems of linear difference equations ................ 189
9 Vector calculus ............................................ 202
9.1 Introduction .......................................... 202
9.2 Partial and total derivatives ......................... 202
9.3 Chain rule and product rule ........................... 207
9.4 Elasticities .......................................... 211
9.5 Directional derivatives and tangent hyperplanes ....... 213
9.6 Taylor's theorem: deterministic version ............... 217
9.7 Multiple integration .................................. 224
9.8 Implicit function theorem ............................. 236
10 Convexity and optimization ................................. 244
10.1 Introduction .......................................... 244
10.2 Convexity and concavity ............................... 244
10.3 Unconstrained optimization ............................ 257
10.4 Equality-constrained optimization ..................... 261
10.5 Inequality-constrained optimization ................... 270
10.6 Duality ............................................... 278
Part II APPLICATIONS Introduction
11 Macroeconomic applications ................................. 289
11.1 Introduction .......................................... 289
11.2 Dynamic linear macroeconomic models ................... 289
11.3 Input-output analysis ................................. 294
12 Single-period choice under certainty ....................... 299
12.1 Introduction .......................................... 299
12.2 Definitions ........................................... 299
12.3 Axioms ................................................ 301
12.4 The consumer's problem and its dual ................... 307
12.5 General equilibrium theory ............................ 316
12.6 Welfare theorems ...................................... 323
13 Probability theory ......................................... 334
13.1 Introduction .......................................... 334
13.2 Sample spaces and random variables .................... 334
13.3 Applications .......................................... 338
13.4 Vector spaces of random variables ..................... 343
13.5 Random vectors ........................................ 345
13.6 Expectations and moments .............................. 347
13.7 Multivariate normal distribution ...................... 351
13.8 Estimation and forecasting ............................ 354
13.9 Taylor's theorem: stochastic version .................. 355
13.10 Jensen's inequality .................................. 356
14 Quadratic programming and econometric applications ......... 371
14.1 Introduction .......................................... 371
14.2 Algebra and geometry of ordinary least squares ........ 371
14.3 Canonical quadratic programming problem ............... 377
14.4 Stochastic difference equations ....................... 382
15 Multi-period choice under certainty ........................ 394
15.1 Introduction .......................................... 394
15.2 Measuring rates of return ............................. 394
15.3 Multi-period general equilibrium ...................... 400
15.4 Term structure of interest rates ...................... 401
16 Single-period choice under uncertainty ..................... 415
16.1 Introduction .......................................... 415
16.2 Motivation ............................................ 415
16.3 Pricing state-contingent claims ....................... 416
16.4 The expected-utility paradigm ......................... 423
16.5 Risk aversion ......................................... 429
16.6 Arbitrage, risk neutrality and the efficient markets
hypothesis ............................................ 434
16.7 Uncovered interest rate parity: Siegel's paradox
revisited ............................................. 436
16.8 Mean-variance paradigm ................................ 440
16.9 Other non-expected-utility approaches ................. 442
17 Portfolio theory ........................................... 448
17.1 Introduction .......................................... 448
17.2 Preliminaries ......................................... 448
17.3 Single-period portfolio choice problem ................ 450
17.4 Mathematics of the portfolio frontier ................. 457
17.5 Market equilibrium and the capital asset pricing
model ................................................. 478
17.6 Multi-currency considerations ......................... 487
Notes ...................................................... 493
References ................................................. 501
Index ...................................................... 505
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