Skip to content
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
  • Privacy Policy
  • 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

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

Enter your email address:

Delivered by FeedBurner

Links

Home

Ken Judd's Personal Site

Ken Judd's Blog

Privacy Policy

  • 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

Copyright © 2025 Numerical Methods in Economics | Powered by Astra WordPress Theme