Essential Mathematics for Economics

Cover
Half Title
Title Page
Copyright Page
Contents
Preface
List of Figures
List of Symbols
List of Important Theorems
Chapter 0: Roadmap
Section I: Introduction to Optimization
	Chapter 1: Existence of Solutions
		1.1. INTRODUCTION
		1.2. THE REAL NUMBER SYSTEM
		1.3. CONVERGENCE OF SEQUENCES
		1.4. THE SPACE RN
		1.5. TOPOLOGY OF RN
		1.6. CONTINUOUS FUNCTIONS
		1.7. EXTREME VALUE THEOREM
		1.A. TOPOLOGICAL SPACE
	Chapter 2: One-variable Optimization
		2.1. INTRODUCTION
		2.2. DIFFERENTIATION
		2.3. NECESSARY CONDITION
		2.4. MEAN VALUE AND TAYLOR’S THEOREM
		2.5. SUFFICIENT CONDITION
		2.6. OPTIMAL SAVINGS PROBLEM
	Chapter 3: Multi-variable Unconstrained Optimization
		3.1. INTRODUCTION
		3.2. LINEAR MAPS AND MATRICES
		3.3. DIFFERENTIATION
		3.4. CHAIN RULE
		3.5. NECESSARY CONDITION
	Chapter 4: Introduction to Constrained Optimization
		4.1. INTRODUCTION
		4.2. ONE LINEAR CONSTRAINT
		4.3. MULTIPLE LINEAR CONSTRAINTS
		4.4. KARUSH-KUHN-TUCKER THEOREM
		4.5. INEQUALITY AND EQUALITY CONSTRAINTS
		4.6. CONSTRAINED MAXIMIZATION
		4.7. DROPPING NONNEGATIVITY CONSTRAINTS
Section II: Matrix and Nonlinear Analysis
	Chapter 5: Vector Space, Matrix, and Determinant
		5.1. INTRODUCTION
		5.2. VECTOR SPACE
		5.3. SOLVING LINEAR EQUATIONS
		5.4. DETERMINANT
	Chapter 6: Spectral Theory
		6.1. INTRODUCTION
		6.2. EIGENVALUE AND EIGENVECTOR
		6.3. DIAGONALIZATION
		6.4. INNER PRODUCT AND NORM
		6.5. UPPER TRIANGULARIZATION
		6.6. POSITIVE DEFINITE MATRICES
		6.7. SECOND-ORDER OPTIMALITY CONDITION
		6.8. MATRIX NORM AND SPECTRAL RADIUS
	Chapter 7: Metric Space and Contraction
		7.1. METRIC SPACE
		7.2. COMPLETENESS AND BANACH SPACE
		7.3. CONTRACTION MAPPING THEOREM
		7.4. BLACKWELL’S SUFFICIENT CONDITION
		7.5. PEROV CONTRACTION
		7.6. PARAMETRIC CONTINUITY OF FIXED POINT
	Chapter 8: Implicit Function and Stable Manifold Theorem
		8.1. INTRODUCTION
		8.2. INVERSE FUNCTION THEOREM
		8.3. IMPLICIT FUNCTION THEOREM
		8.4. OPTIMAL SAVINGS PROBLEM
		8.5. OPTIMAL PORTFOLIO PROBLEM
		8.6. STABLE MANIFOLD THEOREM
		8.7. OVERLAPPING GENERATIONS MODEL
	Chapter 9: Nonnegative Matrices
		9.1. INTRODUCTION
		9.2. MARKOV CHAIN
		9.3. PERRON’S THEOREM
		9.4. IRREDUCIBLE NONNEGATIVE MATRICES
		9.5. METZLER MATRICES
Section III: Convex and Nonlinear Optimization
	Chapter 10: Convex Sets
		10.1. CONVEX SETS
		10.2. CONVEX HULL
		10.3. HYPERPLANES AND HALF SPACES
		10.4. SEPARATION OF CONVEX SETS
		10.5. CONE AND DUAL CONE
		10.6. NO-ARBITRAGE ASSET PRICING
	Chapter 11: Convex Functions
		11.1. CONVEX AND QUASI-CONVEX FUNCTIONS
		11.2. CONVEXITY-PRESERVING OPERATIONS
		11.3. DIFFERENTIAL CHARACTERIZATION
		11.4. CONTINUITY OF CONVEX FUNCTIONS
		11.5. HOMOGENEOUS QUASI-CONVEX FUNCTIONS
		11.6. LOG-CONVEX FUNCTIONS
	Chapter 12: Nonlinear Programming
		12.1. INTRODUCTION
		12.2. NECESSARY CONDITION
		12.3. KARUSH-KUHN-TUCKER THEOREM
		12.4. CONSTRAINT QUALIFICATIONS
		12.5. SADDLE POINT THEOREM
		12.6. DUALITY
		12.7. SUFFICIENT CONDITIONS
		12.8. PARAMETRIC DIFFERENTIABILITY
		12.9. PARAMETRIC CONTINUITY
Section IV: Dynamic Optimization
	Chapter 13: Introduction to Dynamic Programming
		13.1. INTRODUCTION
		13.2. KNAPSACK PROBLEM
		13.3. SHORTEST PATH PROBLEM
		13.4. OPTIMAL SAVINGS PROBLEM
		13.5. OPTIMAL STOPPING PROBLEM
		13.6. SECRETARY PROBLEM
		13.7. ABSTRACT FORMULATION
	Chapter 14: Contraction Methods
		14.1. INTRODUCTION
		14.2. MARKOV DYNAMIC PROGRAM
		14.3. SEQUENTIAL AND RECURSIVE FORMULATIONS
		14.4. PROPERTIES OF VALUE FUNCTION
		14.5. RESTRICTING SPACES
		14.6. STATE-DEPENDENT DISCOUNTING
		14.7. WEIGHTED SUPREMUM NORM
		14.8. NUMERICAL DYNAMIC PROGRAMMING
	Chapter 15: Variational Methods
		15.1. INTRODUCTION
		15.2. EULER EQUATION
		15.3. TRANSVERSALITY CONDITION
		15.4. STOCHASTIC CASE
		15.5. OPTIMAL SAVINGS PROBLEM
Appendix A: Introduction to Numerical Analysis
	A.1. Introduction
	A.2. Solving nonlinear equations
	A.3. Polynomial approximation
	A.4. Quadrature
	A.5. Discretization
Bibliography
Index