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Numerical Methods in Economics
Numerical Methods in Economics
  • 1 Introduction
    • 1.1 What Economists Can Compute
    • 1.2 Roles of Computation in Economic Analysis
    • 1.3 Computation in Science
    • 1.4 The Future of Computing
    • 1.5 The Objectives and Nature of This Book
    • 1.6 Basic Mathematics, Notation, and Terminology
    • 1.7 Software and Supplemental Material
    • 1.8 Further Reading
    • 1.9 Exercises
  • 2 Elementary Concepts
    • 2.1 Computer Arithmetic
    • 2.2 Computer Processing and Algorithms
    • 2.3 The Economics of Computation
    • 2.4 Efficient Polynomial Evaluation
    • 2.5 Efficient Computation of Derivatives
    • 2.6 Direct versus Iterative Methods
    • 2.7 Errors: The Central Problem of Numerical Mathematics
    • 2.8 Making Innite Sequences Finite
    • 2.9 Methods of Approximation
    • 2.10 Evaluating the Errors in the Final Result
    • 2.11 Computational Complexity
    • 2.12 Further Reading and Summary
    • 2.13 Exercises
  • 3 Linear Equations and Iterative Methods
    • 3.1 Gaussian Elimination, LU Decomposition
    • 3.2 Alternative Methods
    • 3.3 Banded Sparse Matrix Methods
    • 3.4 General Sparse Matrix Methods
    • 3.5 Error Analysis
    • 3.6 Iterative Methods
    • 3.7 Operator Splitting Approach
    • 3.8 Convergence of Iterative Schemes
    • 3.9 Acceleration and Stabilization Methods
    • 3.10 Calculating A^−1
    • 3.11 Computing Ergodic Distributions
    • 3.12 Overidentied Systems
    • 3.13 Software
    • 3.14 Summary and Further Reading
    • 3.15 Exercises
  • 4 Optimization
    • 4.1 One-Dimensional Minimization
    • 4.2 Multidimensional Optimization: Comparison Methods
    • 4.3 Newtons Method for Multivariate Problems
    • 4.4 Direction Set Methods
    • 4.5 Nonlinear Least Squares
    • 4.6 Linear Programming
    • 4.7 Constrained Nonlinear Optimization
    • 4.8 Incentive Problems
    • 4.9 Computing Nash Equilibrium
    • 4.10 A Portfolio Problem
    • 4.11 A Simple Econometric Example
    • 4.12 A Dynamic Optimization Problem
    • 4.13 Software
    • 4.14 Further Reading and Summary
    • 4.15 Exercises
  • 5 Nonlinear Equations
    • 5.1 One-Dimensional Problems: Bisection
    • 5.2 One-Dimensional Problems: Newtons Method
    • 5.3 Special Methods for One-Dimensional Problems
    • 5.4 Elementary Methods for Multivariate Nonlinear Equations
    • 5.5 Newtons Method for Multivariate Equations
    • 5.6 Methods That Enhance Global Convergence
    • 5.7 Advantageous Transformations
    • 5.8 A Simple Continuation Method
    • 5.9 Homotopy Continuation Methods
    • 5.10 A Simple CGE Problem
    • 5.11 Software
    • 5.12 Further Reading and Summary
    • 5.13 Exercises
  • 6 Approximation Methods
    • 6.1 Local Approximation Methods
    • 6.2 Ordinary Regression as Approximation
    • 6.3 Orthogonal Polynomials
    • 6.4 Least-Squares Orthogonal Polynomial Approximation
    • 6.5 Uniform Approximation
    • 6.6 Interpolation
    • 6.7 Approximation through Interpolation and Regression
    • 6.8 Piecewise Polynomial Interpolation
    • 6.9 Splines
    • 6.10 Examples
    • 6.11 Shape-Preserving Approximation
    • 6.12 Multidimensional Approximation
    • 6.13 Finite Element Approximations
    • 6.14 Neural Networks
    • 6.15 Further Reading and Summary
    • 6.16 Exercises
  • 7 Numerical Integration and Differentiation
    • 7.1 Newton-Cotes Formulas
    • 7.2 Gaussian Formulas
    • 7.3 Singular Integrals
    • 7.4 Adaptive Quadrature
    • 7.5 Multidimensional Quadrature
    • 7.6 Example: Portfolio Problems
    • 7.7 Numerical Differentiation
    • 7.8 Software
    • 7.9 Further Reading and Summary
    • 7.10 Exercises
  • 8 Monte Carlo and Simulation Methods
    • 8.1 Pseudorandom Number Generation
    • 8.2 Monte Carlo Integration
    • 8.3 Optimization by Stochastic Search
    • 8.4 Stochastic Approximation
    • 8.5 Standard Optimization Methods with Simulated Objectives
    • 8.6 Further Reading and Summary
    • 8.7 Exercises
  • 9 Quasi-Monte Carlo Methods
    • 9.1 Equidistributed Sequences
    • 9.2 Low Discrepancy Methods
    • 9.3 Fourier Analytic Methods
    • 9.4 The Method of Good Lattice Points
    • 9.5 Estimating Quasi-Monte Carlo Errors
    • 9.6 Acceleration Methods and qMC Schemes
    • 9.7 Further Reading and Summary
    • 9.8 Exercises
  • About the Book
  • About the Book – 2nd Edition
    • 5 Nonlinear Equations – 2nd Edition
      • 5.1 One-Dimensional Problems: Bisection 2nd Edition
      • 5.2 One-Dimensional Problems: Newtons Method 2nd Edition
      • 5.3 Special Methods for One-Dimensional Problems 2nd Edition
      • 5.4 Elementary Methods for Multivariate Nonlinear Equations – 2nd Edition
      • 5.5 Newtons Method for Multivariate Equations – 2nd Edition
      • 5.6 Methods That Enhance Global Convergence – 2nd Edition
      • 5.7 Advantageous Transformations – 2nd Edition
      • 5.8 A Simple Continuation Method – 2nd Edition
      • 5.9 Homotopy Continuation Methods – 2nd Edition
      • 5.10 A Simple CGE Problem – 2nd Edition
      • 5.11 Software – 2nd Edition
      • 5.12 Further Reading and Summary – 2nd Edition
      • 5.13 Exercises – 2nd Edition
    • 1 Introduction – 2nd Edition
      • 1.1 What Economists Can Compute – 2nd Edition
      • 1.2 Roles of Computation in Economic Analysis – 2nd Edition
      • 1.3 Computation in Science – 2nd Edition
      • 1.4 The Future of Computing – 2nd Edition
      • 1.5 The Objectives and Nature of This Book — 2nd Edition
      • 1.6 Basic Mathematics, Notation, and Terminology — 2nd Edition
      • 1.7 Software and Supplemental Material – 2nd Edition
      • 1.8 Further Reading — 2nd Edition
      • 1.9 Exercises — 2nd Edition
    • 10 Numerical Quadrature – 2nd Edition
    • 11 Monte Carlo and quasi-Monte Carlo – 2nd Edition
    • 13 Projection Methods for Functional Equations – 2nd Edition
      • 13.1 An Ordinary Differential Equation Example
      • 13.2 A Partial Differential Equation Example – 2nd Edition
      • 13.3 General Projection Method – 2nd Edition
      • 13.4 Boundary Value Problems- 2nd Edition
      • 13.5 Continuous-Time Growth Model- 2nd Edition
      • 13.6 Computing Conditional Expectations – 2nd Edition
      • 13.7 Further Reading and Summary – 2nd Edition
      • 13.8 Exercises – 2nd Edition
    • 13 Projection Methods- 2nd Edition
    • 15 Perturbation Methods in Euclidean Spaces – 2nd Edition
    • 16 Perturbation Methods in Function Spaces – 2nd Edition
    • 17 Asymptotic Methods – 2nd Edition
      • 17.1 Bifurcation Methods – 2nd Edition
      • 17.2 Portfolio Choices for Small Risks – 2nd Edition
      • 17.3 Gauge Functions and Asymptotic Expansions – 2nd Edition
      • 17.4 Method of Undetermined Gauges – 2nd Edition
      • 17.5 Asympotic Expansions of Integrals – 2nd Edition
      • 17.6 Hybrid Perturbation-Projection Methods – 2nd Edition
      • 17.7 Further Reading and Summary – 2nd Edition
      • 17.8 Exercises – 2nd Edition
    • 18 Perfect Foresight Models – 2nd Edition
    • 19 Rational Expectations Models – 2nd Edition
    • 2 Elementary Aspects of Numerical Analysis 2nd Edition
    • 20 Dynamic Games – 2nd Edition
    • 21 Supergames – 2nd Edition
    • 22 Numerical Algebraic Geometry – 2nd Edition
    • 3 Linear Equations – 2nd Edition
    • 4 Unconstrained Optimization – 2nd Edition
    • 6 Constrained Optimization – 2nd Edition
    • 7 Nonlinear Complementarity Problems – 2nd Edition
    • 8 Classical Approximation Methods – 2nd Edition
    • 9 Modern Approximation Methods – 2nd Edition
    • 12 Finite-Difference Methods – 2nd Edition
      • 12.1 Classification of Ordinary Differential Equations – 2nd Edition
      • 12.2 Solution of Linear Dynamic Systems – 2nd Edition
      • 12.3 Finite-Difference Methods for IVPs – 2nd Edition
      • 12.4 Economic Examples of IVPs – 2nd Edition
      • 12.5 Boundary Value Problems for ODEs: Shooting – 2nd Edition
      • 12.6 Finite-Horizon Optimal Control Problems – 2nd Edition
      • 12.7 Innite-Horizon Optimal Control and Shooting – 2nd Edition
      • 12.8 Integral Equations – 2nd Edition
      • 12.9 Further Reading and Summary – 2nd Edition
      • 12.10 Exercises – 2nd Edition
    • 14 Numerical Dynamic Programming — 2nd Edition
      • 14.1 Discrete Time Dynamic Programming Problems – 2nd Edition
      • 14.2 Continuous Time Dynamic Programming Problems – 2nd Edition
      • 14.3 Finite-State Methods — 2nd Edition
      • 14.4 Acceleration Methods for Infinite-Horizon Problems – 2nd Edition
      • 14.5 Discretization Methods for Continuous Problems — 2nd Edition
      • 14.6 Methods for Solving Linear-Quadratic Problems — 2nd Edition
      • 14.7 Continuous Methods for Continuous-State Problems — 2nd Edition
      • 14.8 Parametric Approximations and Simulation Methods — 2nd Edition
      • 14.9 Shape-Preserving Methods — 2nd Edition
      • 14.10 Continuous-Time Problems – 2nd Edition
      • 14.11 Further Readings and Summary – 2nd Edition
      • 14.12 Exercises — 2nd Edition
  • Ken Judd’s Books
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  • 10 Finite-Difference Methods
    • 10.1 Classification of Ordinary Differential Equations
    • 10.2 Solution of Linear Dynamic Systems
    • 10.3 Finite-Difference Methods for IVPs
    • 10.4 Economic Examples of IVPs
    • 10.5 Boundary Value Problems for ODEs: Shooting
    • 10.6 Finite-Horizon Optimal Control Problems
    • 10.7 Innite-Horizon Optimal Control and Shooting
    • 10.8 Integral Equations
    • 10.9 Further Reading and Summary
    • 10.10 Exercises
  • 11 Projection Methods for Functional Equations
    • 11.1 An Ordinary Differential Equation Example
    • 11.2 A Partial Differential Equation Example
    • 11.3 General Projection Method
    • 11.4 Boundary Value Problems
    • 11.5 Continuous-Time Growth Model
    • 11.6 Computing Conditional Expectations
    • 11.7 Further Reading and Summary
    • 11.8 Exercises
  • 12 Numerical Dynamic Programming
    • 12.1 Discrete Time Dynamic Programming Problems
    • 12.2 Continuous Time Dynamic Programming Problems
    • 12.3 Finite-State Methods
    • 12.4 Acceleration Methods for Infinite-Horizon Problems
    • 12.5 Discretization Methods for Continuous Problems
    • 12.6 Methods for Solving Linear-Quadratic Problems
    • 12.7 Continuous Methods for Continuous-State Problems
    • 12.8 Parametric Approximations and Simulation Methods
    • 12.9 Shape-Preserving Methods
    • 12.10 Continuous-Time Problems
    • 12.11 Further Readings and Summary
    • 12.12 Exercises
  • 13 Regular Perturbations of Simple System
    • 13.1 The Mathematics of Regular Perturbation Methods
    • 13.2 Comparative Statics
    • 13.3 Perturbing an IVP
    • 13.4 Perturbing a BVP: Comparative Dynamics
    • 13.5 Continuous-Time, Deterministic Control with One State and One
    • 13.6 Stochastic Control
    • 13.7 Perturbing Discrete-Time Systems
    • 13.8 Perturbing Jump Process Control Problems
    • 13.9 Global Quality Test of Asymptotic Approximation
    • 13.10 Exercises
  • 14 Regular Perturbations in Multidimensional System
    • 14.1 Multidimensional Comparative Statics and Tensor Notation
    • 14.2 Linearization of Multidimensional Dynamic System
    • 14.3 Locally Asymptotically Stable Multidimensional Control
    • 14.4 Perturbations of Discrete-Time Problems
    • 14.5 Multisector, Stochastic Growth
    • 14.6 Further Reading and Summary
    • 14.7 Exercises
  • 15 Advanced Asymptotic Methods
    • 15.1 Bifurcation Methods
    • 15.2 Portfolio Choices for Small Risks
    • 15.3 Gauge Functions and Asymptotic Expansions
    • 15.4 Method of Undetermined Gauges
    • 15.5 Asymptotic Expansions of Integrals
    • 15.6 Hybrid Perturbation-Projection Methods
    • 15.7 Further Reading and Summary
    • 15.8 Exercises
  • 16 Solution Methods for Perfect Foresight Models
    • 16.1 A Simple, Autonomous Overlapping Generations Model
    • 16.2 Equilibrium in OLG Models: Time Domain Methods
    • 16.3 Fair-Taylor Method
    • 16.4 Recursive Models and Dynamic Iteration Methods
    • 16.5 Recursive Models with Nonlinear Equation Methods
    • 16.6 Accuracy Measures
    • 16.7 Tax and Monetary Policy in Dynamic Economies
    • 16.8 Recursive Solution of an OLG Model
    • 16.9 “Consistent” Capital Income Taxation
    • 16.10 Further Reading and Summary
    • 16.11 Exercises
  • 17 Solving Rational Expectations Models
    • 17.1 The Lucas Asset Pricing Model
    • 17.2 Monetary Equilibrium
    • 17.3 Information and Asset Markets
    • 17.4 Commodity Storage Models
    • 17.5 A Simple Stochastic Dynamic Growth Model
    • 17.6 Fixed-Point Iteration
    • 17.7 Time Iteration
    • 17.8 Generalizations
    • 17.9 Further Reading and Summary
    • 17.10 Exercises

3 Linear Equations and Iterative Methods

3.1 Gaussian Elimination, LU Decomposition

3.2 Alternative Methods

3.3 Banded Sparse Matrix Methods 

3.4 General Sparse Matrix Methods

3.5 Error Analysis 

3.6 Iterative Methods 

3.7 Operator Splitting Approach 

3.8 Convergence of Iterative Schemes 

3.9 Acceleration and Stabilization Methods

3.10 Calculating A^−1

3.11 Computing Ergodic Distributions

3.12 Overidentied Systems

3.13 Software

3.14 Summary and Further Reading

3.15 Exercises

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  • Resources by Chapter
    • 1 Introduction
    • 2 Elementary Concepts
    • 3 Linear Equations and Iterative Methods
    • 4 Optimization
    • 5 Nonlinear Equations
    • 6 Approximation Methods
    • 7 Numerical Integration and Differentiation
    • 8 Monte Carlo and Simulation Methods
    • 9 Quasi-Monte Carlo Methods
    • 10 Finite-Difference Methods
    • 11 Projection Methods for Functional Equations
    • 12 Numerical Dynamic Programming
    • 13 Regular Perturbations of Simple System
    • 14 Regular Perturbations in Multidimensional System
    • 15 Advanced Asymptotic Methods
    • 16 Solution Methods for Perfect Foresight Models
    • 17 Solving Rational Expectations Models

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