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This book presents a coherent and systematic exposition of the mathematical theory of the problems of optimization and stability. Both of these are topics central to economic analysis since the latter is so much concerned with the optimizing behaviour of economic agents and the stability of the interaction processes to which this gives rise. The topics covered include convexity, mathematical programming, fixed point theorems, comparative static analysis and duality, the stability of dynamic systems, the calculus of variations and optimal control theory. The authors present a more detailed and wide-ranging discussion of these topics than is to be found in the few books which attempt a similar coverage. Although the text deals with fairly advanced material, the mathematical prerequisites are minimised by the inclusion of an integrated mathematical review designed to make the text self-contained and accessible to the reader with only an elementary knowledge of calculus and linear algebra. A novel feature of the book is that it provides the reader with an understanding and feel for the kinds of mathematical techniques most useful for dealing with particular economic problems. This is achieved through an extensive use of a broad range of economic examples (rather than the numerical/algebraic examples so often found).This is suitable for use in advanced undergraduate and postgraduate courses in economic analysis and should in addition prove a useful reference work for practising economists.

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CONTENTS
Preface = xiii
1 CONVEXITY = 1
  1.1 Introduction = 1
  1.2 Convex sets = 1
    1.2.1 Basic properties = 1
    1.2.2 Separation of convex sets = 6
  1.3 Convex and concave functions = 16
    1.3.1 Basic properties = 16
    1.3.2 Quasi-concave functions = 25
    1.3.3 Extremum properties = 28
2 STATIC OPTIMIZATION = 32
  2.1 Introduction - 32
  2.2 Classical optimization = 34
    2.2.1 Unconstrained optimization = 34
    2.2.2 Equality constrained optimization = 37
  2.3 Non-linear programming = 48
    2.3.1 Saddlepoint optimization = 48
    2.3.2 Constraint qualifications = 57
    2.3.3 Quasiconcave programming = 68
3 EQUILIBRIUM MATHEMATICS = 73
  3.1 Introduction = 73
  3.2 Equilibrium mathematics = 73
    3.2.1 Continuity of correspondences = 74
    3.2.2 Fixed-point theorems = 84
  3.3 Existence of competitive equilibrium = 87
    3.3.1 Introduction = 87
    3.3.2 Finite private ownership economy = 88
    3.3.3 Properties of demands and supplies = 89
    3.3.4 Equilibrium = 92
  3.4 Equilibrium in n-person non-cooperative games = 94
4 COMPARATIVE STATICS AND DUALITY = 98
  4.1 Introduction = 98
  4.2 Comparative statics and maximum value functions = 99
  4.3 An introduction to duality theory = 117
5 DYNAMICS AND STABILITY = 134
  5.1 Introduction = 134
  5.2 Differential equations : basic concepts = 135
    5.2.1 Existence and uniqueness theorems = 135
    5.2.2 Equilibrium and stability concepts = 141
  5.3 Linear differential and stability concepts = 144
  5.4 Stability of linear differential equations = 151
    5.4.1 Necessary and sufficient conditions for stability = 151
    5.4.2 Necessary conditions for stability = 154
    5.4.3 Sufficient conditions for stability = 154
  5.5 Trajectories, conditional stability and saddle point equilibria = 155
    5.5.1 Trajectories = 155
    5.5.2 The stable manifold = 156
    5.5.3 Conditional stability = 159
    5.5.4 Identifying the stable manifold = 160
  5.6 Stability analysis for non-linear systems = 162
    5.6.1 The linear approximation method = 163
    5.6.2 The Olech sufficient conditions for global stability = 166
    5.6.3 Liapunov's second method = 167
    5.6.4 Non-unique equilibria = 176
    5.6.5 Phase diagrams = 179
  5.7 Economic applications = 185
    5.7.1 A resource market model = 185
    5.7.2 Stability of general equilibrium = 189
    5.7.3 A partly rational expectations macro model = 198
  5.8 Dynamical systems : topological considerations = 203
    5.8.1 Dynamical systems : introduction = 203
    5.8.2 The system x · ＝f(x), x ∈ X ⊂ R 2 = 206
    5.8.3 The system x · ＝f(x), x ∈ X ⊂ R 2 = 223
  5.9 Difference equation systems = 226
    5.9.1 Introduction = 226
    5.9.2 The linear difference equation system = 228
    5.9.3 Equilibrium and stability for linear systems = 229
6 INTRODUCTION TO DYNAMIC OPTIMIZATION AND THE CALCULUS OF VARIATIONS = 233
  6.1 Introduction = 223
  6.2 Mathematical preliminaries = 238
  6.3 Separability and the principle of optimality = 240
  6.4 Calculus of variations : necessary conditions = 241
    6.4.1 First order necessary conditions = 242
    6.4.2 Second order necessary conditions = 250
    6.4.3 Weierstrass necessary conditions = 251
  6.5 Variable endpoints and transversality conditions = 252
  6.6 The bequest or final value function = 259
  6.7 Corners = 260
  6.8 Summary of results for x(t) ∈ R n = 262
    6.8.1 Fixed endpoint problem = 263
    6.8.2 Variable endpoint problem = 264
  6.9 Sufficient conditions for finite horizon problems = 265
  6.10 Constraints in the calculus of variations = 267
    6.10.1 Global, integral constrains = 267
    6.10.2 Local constraints = 269
  6.11 An extended example = 272
  6.12 Infinite horizon problems = 279
7 OPTIMAL CONTROL THEORY = 288
  7.1 Introduction = 288
    7.1.1 A variational approach to necessary conditions = 288
    7.1.2 Transversality conditions = 293
    7.1.3 Interpretation of the multipliers = 294
  7.2 Necessary conditions = 296
  7.3 Pontryagin's maximum principle = 305
    7.3.1 The control problem = 305
    7.3.2 Behaviour of the co-state variables = 308
    7.3.3 Spatial variations = 310
  7.4 Constraints on state and control variables = 315
  7.5 Sufficient conditions : finite horizon problems = 327
  7.6 Bounded state variable constraints = 330
  7.7 An extended example = 335
  7.8 Infinite horizon problems = 346
APPENDICES - MATHEMATICAL REVIEW = 358
  Sets and mapping = 358
  Real and complex number fields = 361
  Vector spaces = 364
  Matrices = 368
  Linear equations = 374
  Eigenvalues = 374
  Quadratic forms = 378
  Further set properties = 380
  Continuity concepts = 382
  Differentiability = 388
  Taylor's theorem = 393
  Homogeneous functions = 394
  Implicit function theorem = 394
  Integration = 395
  Glossary = 399
  Bibliography = 401
  Author Index = 408
  Subject Index = 410