Alexis Akira Toda
Essential Mathematics for Economics
Alexis Akira Toda
Essential Mathematics for Economics
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This book covers mathematical topics that are essential for economic analysis in a concise but rigorous fashion. The book covers selected topics such as linear algebra, real analysis, convex analysis, constrained optimization, dynamic programming, and numerical analysis in a single volume.
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This book covers mathematical topics that are essential for economic analysis in a concise but rigorous fashion. The book covers selected topics such as linear algebra, real analysis, convex analysis, constrained optimization, dynamic programming, and numerical analysis in a single volume.
Produktdetails
- Produktdetails
- Verlag: Taylor & Francis Ltd
- Seitenzahl: 320
- Erscheinungstermin: 18. Oktober 2024
- Englisch
- Abmessung: 234mm x 156mm
- ISBN-13: 9781032698946
- ISBN-10: 1032698942
- Artikelnr.: 70527771
- Verlag: Taylor & Francis Ltd
- Seitenzahl: 320
- Erscheinungstermin: 18. Oktober 2024
- Englisch
- Abmessung: 234mm x 156mm
- ISBN-13: 9781032698946
- ISBN-10: 1032698942
- Artikelnr.: 70527771
Alexis Akira Toda was born in Montreal, Canada to a Japanese father and a French-Canadian mother and moved to Japan at the age of four. He received a B.A. in medicine from University of Tokyo in 2004 and then an M.A. in economics in 2008 while practicing anesthesia. He moved to the United States in 2008 and obtained a Ph.D. in economics from Yale University in 2013. After teaching mathematics for economics, mathematical economics, and finance at UC San Diego for eleven years, he moved to Emory University in 2024, where he is a professor of economics. He is the author of more than 40 research articles published in economics, mathematics, physics, and medical journals on a wide variety of topics including general equilibrium, macro-finance, consumption and savings, income and wealth distributions, asset price bubbles, power laws, dynamic programming, econometrics, and numerical methods, among others. He can be reached at https://alexisakira.github.io.
0. Roadmap. Section I. Introduction to Optimization. 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. 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. 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. 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. 5. Vector Space, Matrix, and Determinant. 5.1. Introduction. 5.2.
Vector Space. 5.3. Solving Linear Equations. 5.4. Determinant. 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. 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. 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. 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. 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. 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. 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. 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. 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. 15.
Variational Methods. 15.1. Introduction. 15.2. Euler Equation. 15.3.
Transversality Condition. 15.4. Stochastic Case. 15.5. Optimal Savings
Problem.
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. 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. 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. 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. 5. Vector Space, Matrix, and Determinant. 5.1. Introduction. 5.2.
Vector Space. 5.3. Solving Linear Equations. 5.4. Determinant. 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. 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. 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. 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. 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. 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. 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. 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. 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. 15.
Variational Methods. 15.1. Introduction. 15.2. Euler Equation. 15.3.
Transversality Condition. 15.4. Stochastic Case. 15.5. Optimal Savings
Problem.
0. Roadmap. Section I. Introduction to Optimization. 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. 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. 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. 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. 5. Vector Space, Matrix, and Determinant. 5.1. Introduction. 5.2.
Vector Space. 5.3. Solving Linear Equations. 5.4. Determinant. 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. 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. 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. 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. 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. 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. 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. 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. 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. 15.
Variational Methods. 15.1. Introduction. 15.2. Euler Equation. 15.3.
Transversality Condition. 15.4. Stochastic Case. 15.5. Optimal Savings
Problem.
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. 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. 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. 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. 5. Vector Space, Matrix, and Determinant. 5.1. Introduction. 5.2.
Vector Space. 5.3. Solving Linear Equations. 5.4. Determinant. 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. 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. 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. 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. 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. 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. 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. 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. 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. 15.
Variational Methods. 15.1. Introduction. 15.2. Euler Equation. 15.3.
Transversality Condition. 15.4. Stochastic Case. 15.5. Optimal Savings
Problem.